Portfolio management in a stochastic factor model under the existence of private information

Portfolio management in a stochastic factor model under the existence of private information Abstract In the present article, we study an optimal control problem for a general stochastic factor model under the existence of private information. More precisely, we consider a portfolio manager who has the possibility to invest part of her wealth in a financial market consisting of a risk-free asset and a risky one, whose coefficients depend on some external stochastic factor. Moreover, we assume that the manager, from the beginning of the trading interval, observes an information signal associated with the future evolution of the risky asset. This information is not clear but is subject to some observation noise. Within a very general framework, by resorting to a mixture of dynamic programming and initial enlargement of filtrations techniques, we characterize the optimal value function and the feedback control law, by solving an expected utility maximization problem under the enlarged information set of the economic agent. In the case of the exponential utility function and considering a specific form for the market parameters and the information signal, we provide closed form solutions for the optimal investment decision and the optimal value function. Additionally, by employing an Euler–Maruyama scheme followed by a Monte-Carlo approach, we numerically study the impact of the private information on the optimal investment strategy for this concrete example. The article has Supplementary Material, which provides the extension of our model to other classes of utility functions (logarithmic and power) and also presents the general case of multiple assets and factors. 1. Introduction Stochastic optimal control theory has found itself to establish a ubiquitous presence in many fields such as mathematical economics and finance. It is the backbone of most of the modern theory of portfolio management, policy design in macroeconomics, international banking or environmental economics. While heavily used in applications, it is also a theory very rich from the mathematical point of view, which still presents very important open problems. To the best of our knowledge, the implementation of stochastic optimal control techniques on financial related problems was originated in the pioneering works of Merton (1969, 1971), who studied a continuous time investment and consumption problem in a financial market consisting of two assets: a risk-free asset and a risky one whose dynamics are described by the geometric Brownian motion model. For the special case where the economic agent operates under constant relative risk aversion preferences, Merton (for more information, see Merton, 1969) was able to provide closed form solutions for the value function, the optimal investment strategy and the optimal consumption rules. Since then, many papers and books that bridge stochastic optimal control techniques and mathematical finance have been written (see, among others, Browne, 1995; Duffie et al., 1997; Zariphopoulou, 2001, 2009; Pham, 2009 and references therein). One of the most salient features of the mentioned theory, is the assumption of a common public information flow, according to which all the economic agents that operate in a financial market make their portfolio (investment and/or consumption) decisions. This information is expressed in terms of the natural filtration which is generated by the stochastic process implying uncertainty in the market, for example a Brownian motion, and is available at everyone’s disposal from the very beginning of the trading interval. This is achieved by postulating that all portfolio strategies are non-anticipating, that is, are progressively measurable with respect to this filtration. However, research and empirical results have provided evidence that this assumption is not fully valid. Niederhoffer & Osborne (1966) pointed out that there exist specialists operating on major security exchanges that have access to some private information concerning the unexecuted limit orders and this information can be used for certain profit purposes. In this vein, Fama (1970) empirically proved that financial markets are not strongly efficient in the sense that private information is not fully reflected in security prices. Ipso facto, the existence of asymmetrically informed economic agents in financial markets is broadly accepted and modelling their behaviour has been one of the most challenging concepts in mathematical finance. A very important and more general class of models of asymmetric information is the inside information models, according to which in a financial market operate two type of agents. The honest economic agents (or regulars) whose portfolio decisions are based on the public information flow which is induced by the market noises and the informed economic agents (or insiders) who exclusively possess some future information concerning, for example, the evolution of the market price movements in addition to the public information flow. Depending when the additional private information is available at the insider’s disposal, there are two sophisticated techniques to effectively treat inside information models, namely, initial (when the additional information is available from the very beginning of the trading interval) and progressive (when the private information is progressively known, that is, as time passes) enlargement of filtrations. By enlargement of a filtration we mean the expansion of a given filtration $$\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$$ to a new filtration $$\mathbb{G}=(\mathcal{G}_{t})_{t\geq 0}$$, defined so as to satisfy the usual hypotheses of right continuity and completeness and $$\mathcal{F}_{t}\subset\mathcal{G}_{t}$$ for each $$t\geq 0$$. According to initial enlargement of filtrations theory, the additional information is stored in a random variable and is available at the insider’s disposal from the beginning of the trading interval. The natural filtration of the market is then enlarged with the sigma algebra which is generated by this random variable and results to the filtration of the insider. This technique has proven itself to be a very efficient tool for the study of the problem of inside information in a variety of cases (see for example, Pikovsky & Karatzas, 1996; Amendinger et al., 1998; Grorud & Pontier, 1998; Imkeller et al., 2001; Fei & Wu, 2003; Amendinger & Imkeller, 2007; Danilova et al., 2010; Kohatsu-Higa & Ortiz-Latorre, 2010; Liu et al., 2010; Hansen, 2013), for certain problems arising in mathematical finance and also (Baltas et al., 2012; Xiong et al., 2014) for an inside information model in the insurance/reinsurance market. In mathematical finance, a sound mathematical model which has been heavily applied to describe the evolution of a stock price process, is the geometric Brownian motion model. This model has its origins in the inspiring work of Samuelson (1965) and acted as point of reference for the celebrated, Nobel prize awarded, Black–Scholes model. Since then, this model has risen to fame and has been extensively studied in many works (see e.g. Shreve & Soner, 1994; Browne, 1995; Duffie et al., 1997 and references therein). However, it fails to capture the effect of many important exogenous market parameters. To the best of our knowledge, the first attempt in extending this model, is the article of Karatzas et al. (1991) who assume the coefficients to be deterministic functions of time. A perhaps more realistic and much general framework is provided by the so called stochastic factor models. Substantially, the main characteristic of such a model is that the prices of the underlying risky asset are allowed to be functions of an another stochastic process, called the stochastic factor, which evolves according to a correlated diffusion process. This general setting covers, among others, the interesting case of stochastic volatility and stochastic interest rate models. Within this framework, there exist many papers and books of both mathematical and practical interest (see, among others, Kim & Omberg, 1996; Fouque et al., 2000; Bielecki & Pliska, 2001; Zariphopoulou, 2001, 2009; Liu, 2007; Delong & Klüppelberg, 2008 and references therein). This work aims to contribute to the existing theory of portfolio management, in complex environments, under the existence of private information. The novelty of our work lies in the fact that, for the very first time in the relative literature, we expand the setting of general stochastic factor models to include the effect of private information in the study of a related optimal investment problem. To be more precise, we consider a portfolio manager who is endowed with some initial wealth and is allowed to invest in a financial market consisting of a risk-free asset (bond or bank account) and a risky one, whose evolution depends on some exogenous economic factor. This factor, evolves stochastically according to a correlated diffusion process with general coefficients. Additionally, the manager possesses, from the very beginning of the trading interval, some additional information concerning the future, which stems from the observation of a private signal, thus introducing in this way inside information aspects to our model. However, this information is not precise, in the sense that is subject to some observation noise. The main focus of interest is to study a general expected utility maximization problem by resorting to a rigorously defined mixture of dynamic programming and initial enlargement of filtrations technique so as to reduce the calculation of the optimal portfolio decision to the solution of a relevant Hamilton–Jacobi–Bellman equation valued for a wide class of utility functions. In this vein, we provide a general result characterizing the value function of the problem at hand and the optimal control law. In the special case when the economic agent operates under exponential preferences, we provide closed form solutions for the value function and the optimal investment decision. In addition, we present a fully worked example by considering a very specific form for the additional information and the market parameters, namely, when the stochastic factor evolves according to an Ornstein–Uhlenbeck process and the information signal concerns a future value of the underlying Brownian motion, driving uncertainty in the risky asset. By employing a mixture of an Euler–Maruyama scheme and a Monte-Carlo approach, we are able to effectively study numerically, in full detail and for the whole trading interval (this is, to the best of our knowledge, among the first papers to do so), the optimal investment strategy of the insider capturing in this way the qualitative and quantitative nature of her behaviour. Additionally, in the associated Supplementary Material, we extend our framework in two major ways: (i) we solve the proposed model in the case of the logarithmic and the power utility functions and (ii) we present the multi-dimensional case with multiple risky assets and stochastic factors. The present article is organized as follows. In Section 2 we describe our model, and in Section 3 we formulate the problem of maximizing the insider’s expected utility from terminal wealth and give a general result characterizing the value function and the optimal investment strategy, by solving the corresponding Hamilton–Jacobi–Bellman equation. Section 4 provides closed form solutions for the value function and the optimal investment strategy, when the economic agent operates under exponential preferences and additionally we provide a fully worked example by considering a specific form for the private information and the market parameters. Finally, in Section 5, we numerically study the results of the specific example of Section 4. The article has Supplementary Material that extends our results by considering (i) other utility functions (logarithmic and power type preferences) and (ii) the case of multiple assets and factors. 2. The model Let us consider the filtered probability space $$(\Omega, \mathscr{F}_{t}, \mathbb{P})$$ that satisfies the usual hypotheses of right continuity and completeness, where $$\mathscr{F}_{t}=\sigma(W(s), B(s)\,, s\leq t)$$ is the natural filtration induced by the standard independent Brownian motions $$\left\{W(t), \, t\geq 0\right\}$$ and $$\left\{B(t), \, t\geq 0\right\}$$. 2.1. The financial market We adopt a model for the financial market on the fixed finite time horizon $$[0,T]$$, with $$T\in(0,\infty)$$, consisting of a risk-free asset (bond or bank account) with dynamics described by \begin{equation} \begin{aligned} dP(t) &= rP(t)dt \\ P(0) &=1, \end{aligned} \end{equation} (2.1) where $$r>0$$ denotes the interest rate and a risky one (stock or index) with dynamics given by \begin{equation} \begin{aligned} \frac{dS(t)}{S(t)}&= \Big[r +\mu(Y(t))\Big]dt + \sigma(Y(t))dW(t)\\ S(0)&=S_{0}>0, \end{aligned} \end{equation} (2.2) where $$P(t)$$ denotes the price of the bond and $$S(t)$$ the price of the stock at time $$t\in[0,T]$$. The stochastic process $$\left\{Y(t),\, t\geq 0\right\}$$ may be interpreted as an external economic factor that has an impact on the prices of the risky asset and satisfies the following stochastic differential equation \begin{equation} \begin{aligned} dY(t)&=\alpha(Y(t))dt + \beta(Y(t))\Big[\rho dW(t) + \sqrt{1-\rho^{2}}dB(t)\Big]\\ Y(0)&=y \in\mathbb{R}, \end{aligned} \end{equation} (2.3) where $$\rho\in[-1,1]$$ stands for the correlation coefficient. We assume that the stochastic factor cannot be directly traded leading to the incompleteness of the market, (as long as $$\rho^{2}\neq 1$$), since the economic agent faces more sources of uncertainty than traded assets. This very general framework has been heavily studied in the relative literature (see e.g. Zariphopoulou, 2009) and allows for the effective modelling of the influence that many important exogenous factors, of either macroeconomic or microeconomics nature, have, on the evolution of the stock prices. Assumption 2.1 The market coefficients $$\ell=(\mu,\sigma,\alpha,\beta)$$ in equations (2.2) and (2.3), satisfy the following conditions (i) The function $$\ell$$ is sublinear and globally Lipschitz, that is, for every $$y,\bar{y}\in\mathbb{R}$$ there exists some constant $$C>0$$, such that \begin{equation*} |\ell(y)-\ell(\bar{y})|\leq C|y-\bar{y}|, \end{equation*} and \begin{equation*} \ell^{2}(y)\leq C(1+y^{2}). \end{equation*} (ii) For every $$y\in\mathbb{R}$$ there exist constants $$c_{1}$$ and $$c_{2}$$, such that \begin{align*} \sigma(y)\geq c_{1}>0,\quad {\rm and}\quad \beta(y)\geq c_{2}>0. \end{align*} (iii) The solution of the stochastic differential equation (2.2) satisfies $$S(t)\geq 0$$ for every $$t\in[0,T]$$ and initial condition $$S(0)>0$$. Remark 2.1 The conditions of Assumption 2.1 are somewhat standard in the relative literature (see e.g. Zariphopoulou, 2001, 2009) and can be relaxed in a number of ways depending on the underlying framework. In our context, it is straightforward to verify that under Assumption 2.1(i) the system of stochastic differential equations (2.2) and (2.3) admits a unique strong solution. For more information on this subject we refer the interested reader to Chapter VI of Arnold (1974). The non-degeneracy Assumption 2.1(ii) guarantees the well-posedness of the associating value function (cf. equation (3.1)) and of the resulting partial differential equation (cf. equation (3.3)). 2.2. Stochastic differential equation for the manager’s wealth We envision a portfolio manager, who, at time $$t=0$$, is endowed with some initial wealth $$x>0$$ and whose actions cannot affect the market prices.1 The portfolio process $$\pi(t)=\pi(t,\omega):[0,T]\times\Omega\rightarrow \Pi\subset\mathbb{R}$$ denotes the proportion of her wealth $$X^{\pi}(t)$$ invested in the risky asset. The remaining proportion $$(1-\pi(t))X^{\pi}(t)$$ is invested in the risk-free asset. Here, $$\Pi$$ is a fixed closed and convex subset of $$\mathbb{R}$$; typically compact. As a result, the wealth process corresponding to the strategy $$\pi(t)$$, is defined as the solution of the following stochastic differential equation \begin{equation*} dX^{\pi}(t) = \pi(t)X^{\pi}(t)\frac{dS(t)}{S(t)} + \left(1-\pi(t)\right)X^{\pi}(t)\frac{dP(t)}{P(t)}. \end{equation*} Therefore, \begin{equation} \begin{aligned} \frac{dX^{\pi}(t)}{X^{\pi}(t)} &= \left[r + \pi(t)\mu(Y(t))\right]dt + \pi(t)\sigma(Y(t))dW(t) \\ X^{\pi}(0)&=x>0. \end{aligned} \end{equation} (2.4) Definition 2.1 Let $$\mathbb{F}$$ be a general filtration. We denote by $$\mathcal{A}({\mathbb{F}};T)$$ the class of admissible strategies $$\pi(t)$$ that satisfy the following conditions: (i) $$\pi(t):[0,T]\times\Omega\rightarrow \Pi\subset\mathbb{R}$$ is a progressively measurable mapping with respect to the filtration $$\mathbb{F}$$; (ii) $$\mathbb{E}\left[\displaystyle\int_{0}^{T} \left(\sigma(Y(t))\pi(t)\right)^{2}dt \right]< \infty,\,\, \mathbb{P}$$-a.s.; (iii) The SDE (2.4) admits a unique strong solution, denoted by $$X^{\pi}(t)$$. 2.3. Inside information Suppose now that the portfolio manager, in addition to the publicly available information flow $$\mathbb{F}=(\mathscr{F}_{t})_{t\in[0,T]}$$, possesses, from the very beginning of the trading interval $$[0,T]$$, some private information through the observation of a Brownian signal, concerning the stochastic process $$\left\{W(t),\, t\geq 0\right\}$$ introducing uncertainty in the stock price process (2.2), at the future time $$T_{0}\geq T$$. This additional information is stored in a certain $$\mathscr{F}_{T_{0}}$$-measurable random variable $$\psi$$, which is available only at her disposal and hidden from any other economic agents that presumably operate on the same financial market. Consequently, the informed portfolio manager (also called insider) can choose her strategy from the class of $$\mathbb{G}$$ adapted processes, where the filtration $$\mathbb{G}=(\mathscr{G}_{t})_{t\in[0,T]}$$ is given by the augmentation of the natural filtration (which is available at everyone’s disposal) with the inside information, represented by the $$\sigma$$-algebra generated by $$\psi$$, that is, \begin{equation*} \begin{aligned} \mathscr{G}_{t} &= \mathscr{F}_{t} \vee \sigma(\psi) \\ &= \sigma(W(s), B(s)\,, s\leq t) \vee \sigma(\psi),\,\, t\in[0,T], \end{aligned} \end{equation*} and is defined so as to satisfy the usual hypotheses of right continuity and completeness. Obviously, $$\mathbb{F} \subset \mathbb{G}$$, hence the insider can choose her portfolio from a larger class of admissible strategies than any other economic agent. Typically, the core of initial enlargement of filtrations theory, dictates, that if the random variable $$\psi$$ satisfies the famous Jacod’s Hypothesis2 (see e.g. Jacod, 1985, Chapter V of Jeanblanc et al., 2009 and Kohatsu-Higa & Ortiz-Latorre, 2010 or Chapter I of Mansuy & Yor, 2006), then, $$W$$ becomes a $$(\mathbb{G}, \mathbb{P})$$-semimartingale with respect to the enlarged filtration. In fact, we have the following well-known result, a version of which can be found in Danilova et al. (2010). Lemma 2.1 Suppose $$\psi$$ is an $$\mathscr{F}$$-measurable random variable, such that it admits a conditional density $$g:[0,T]\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{+}$$ with respect to $$\mathbb{F}$$, so that for any test function $$h:\mathbb{R}\rightarrow\mathbb{R}$$ \begin{equation} \mathbb{E}\Big[h(\psi)|\mathscr{F}_{t}\Big] = \int_{\mathbb{R}}h(r)g(t,r,W(t))dr,\,\, t\in[0,T]. \end{equation} (2.5) Suppose further that $$g(t,r,z)$$ is smooth enough in $$z$$ and it satisfies \begin{equation} \int_{0}^{t}\Big|\frac{g_{z}(s,r,z)}{g(s,r,z)}\Big|ds<\infty\,\,\,\,\text{and}\,\,\,\,\int_{\mathbb{R}}\big|g_{z}(t,r,z)\big|dr<\infty, \end{equation} (2.6) for a.e. $$z\in\mathbb{R}$$ and $$t\in[0,T]$$. Then, the semimartingale decomposition of $$W$$ with respect to $$\mathbb{G}$$ is given by \begin{equation} W(t) = \widetilde{W}(t) + \int_{0}^{t}\frac{g_{z}(s,\psi,W(s))}{g(s,\psi,W(s))}ds,\,\,\,t\in[0,T], \end{equation} (2.7) where $$\widetilde{W}$$ is a $$(\mathbb{G}, \mathbb{P})$$-Brownian motion. The stochastic process \begin{equation} \frac{g_{z}(t,\psi,W(t))}{g(t,\psi,W(t))},\,\,\,t\in[0,T], \end{equation} (2.8) is referred to as the information drift. Proof. For a detailed proof, see e.g. Danilova et al. (2010) and Chapter I of Mansuy & Yor (2006). □ In the present article, we assume that knowledge of the value of $$\psi$$ allows the manager to infer information concerning $$W(T_{0})$$, which however is not explicitly specified at this point. This additional information will serve as a means to predict the movement of stock prices and will possibly help her to obtain a better position in the market described by equations (2.1–2.3). A measure of the efficiency of $$\psi$$ will be denoted by a scalar parameter $$\lambda\in[0,1]$$. To wit, a value of $$\lambda=1$$ implies that the information signal is precise whereas a value of $$\lambda=0$$ implies a completely imprecise information signal.3 For various forms of the inside information, Lemma 2.1 gives the semimartingale decomposition of the stochastic process $$W$$ under the enlarged filtration $$\mathbb{G}$$. As in general the conditional density $$g$$ corresponding to $$\psi$$ is not known, motivated by Lemma 2.1, we adopt the following semimartingale decomposition for $$W$$ under the enlarged filtration $$\mathbb{G}$$, \begin{equation} W(t) = \widetilde{W}(t) + \int_{0}^{t}m(s)ds,\,\,\,t\in[0,T], \end{equation} (2.9) where the stochastic process $$m(t):=m(t,W(t);T_{0},\psi,\lambda)$$ stands for the information drift. Such a choice is not irrational, as equation (2.8) already dictates that the information drift is a stochastic process that depends on the information signal $$\psi$$ and the stochastic process $$W(t)$$. Additionally, due to the assumption on the observed signal, we have to take into account two more parameters associated with the signal, namely $$\lambda$$ and $$T_{0}$$. At this point, and before proceeding any further in our analysis, it is crucial to make the following assumption, as it partially guarantees the applicability of the classical theory (see e.g. Karatzas, 1989; Karatzas & Shreve, 1998). Assumption 2.2 The information drift satisfies the integrability condition \begin{equation*} \mathbb{E}\left[\int_{0}^{T}m^{2}(s)ds\right]<\infty. \end{equation*} Clearly, from equation (2.9) it follows that the stochastic process $$m(t)$$ must be $$\mathscr{G}_{t}$$-measurable. However, in order to make further progress to the stochastic optimal control problem we have in mind, we have to restrict to more specific forms for the information drift. For particular classes of initial enlargements related to Brownian signals, it can be shown (cf. Proposition 4.1 and especially equation (4.11) in view of equation (4.10) and equation (4.24) in view of equation (4.23), see also Danilova et al., 2010; Hansen, 2013) that a plausible assumption for the dynamics of $$m(t)$$ is the following. Assumption 2.3 The information drift evolves according to the following stochastic differential equation \begin{equation} \begin{aligned} dm(t) &= f(t)d\widetilde{W}(t) \\ m(0) &=m_{0}(T_{0},\psi,\lambda)\in\mathbb{R}, \end{aligned} \end{equation} (2.10) where $$f(t):=f(t;T_{0},\lambda)$$ is a square integrable deterministic function, for every $$t\in[0,T]$$. Remark 2.2 At this point, let us comment a little bit the form of the functions appearing in equations (2.9) and (2.10): (i) $$m(t)$$ is the information drift which is generated by the information signal $$\psi$$. This information concerns the $$\left(\mathbb{F}, \mathbb{P}\right)$$-Brownian motion $$W$$, at the future time $$T_{0}$$. Furthermore, it is linked with a scalar parameter $$\lambda$$, denoting its precision level. Hence, it seems natural to let the information drift be a function of $$W$$ and depend on the parameters $$T_{0}, \lambda$$ and $$\psi$$4. (ii) $$f(t)$$ aims to capture the dynamics of the information drift in the enlarged filtration. For the special cases we have in mind, it is obtained by an application of Itô’s lemma to the information drift (assuming, of course, enough smoothness). This approach suggests that $$f(t)$$ takes into account the information horizon $$T_{0}$$ and the parameter $$\lambda$$, and not $$W$$ or $$\psi$$ (iii) $$m(0)$$ is the value of the information drift at time $$t=0$$. Assuming that $$W$$ is a standard Brownian motion (i.e. $$W(0)=0$$), it is natural to expect $$m(0)$$ to be a function of $$T_{0}, \lambda$$ and $$\psi$$. Our intention from now on is to plug the semimartingale decomposition (2.9), by taking into account equation (2.10), in the stochastic differential equation (2.4) that describes the manager’s wealth at time $$t$$, so as to reflect the additional information she possesses. Proposition 2.1 The wealth of the informed economic agent at time $$t\in[0,T]$$, taking explicitly into account the information signal $$\psi$$ she observes at time $$t=0$$, is given by the system of stochastic differential equations \begin{align} \begin{split} \frac{dX^{\pi}(t)}{X^{\pi}(t)}&=\Big[r + \pi(t)\mu(Y(t)) + \pi(t)\sigma(Y(t))m(t)\Big]dt + \pi(t)\sigma(Y(t))d\widetilde{W}(t)\\ dY(t)&= \Big[\alpha(Y(t)) + \rho\beta(Y(t))m(t)\Big]dt + \beta(Y(t))\Big[\rho d\widetilde{W}(t) + \sqrt{1-\rho^{2}}dB(t)\Big]\\ dm(t)&= f(t) d\widetilde{W}(t), \end{split} \end{align} (2.11) with initial condition $$(X^{\pi}(0), Y(0), m(0))=(x,y,m)\in\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}.$$ Proof. The proof follows by substituting equation (2.9) in the stochastic differential equation (2.4) that describes the wealth of the economic agent under symmetric information and in the stochastic differential equation (2.3) that describes the evolution of the external factor process. □ 3. The optimization problem In this section, we formally state the problem of maximizing the expected utility from terminal wealth for the informed economic agent by taking explicitly into account the additional information she possesses. 3.1. The problem We are now in a position to rigorously state the problem. Let $$\mathcal{A}({\mathbb{G}};T)$$ denote the admissible class of the insider taking into account her additional information, as it follows in terms of Definition 2.1. Assumption 3.1 The utility function $$U:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$$ is a function that is strictly increasing, strictly concave, $$C^{1}$$, and satisfies the Inada conditions \begin{align} \lim_{x\rightarrow +\infty}U^{'}(x) &= 0 \nonumber\\ \lim_{x\rightarrow 0}U^{'}(x) &= +\infty \, \Big(\text{or} \lim_{x\rightarrow -\infty}U^{'}(x) = +\infty\Big).\nonumber \end{align} Remark 3.1 Our aim is to enlarge the state space with two additional variables. The variable $$y$$ which corresponds to the stochastic factor process, and the variable $$m$$ which corresponds to the private information. However, one should emphasize that: (i) $$m(t)$$ is adapted to the observed (enlarged) filtration $$\mathbb{G}$$, hence $$m(t)$$ is known to the informed agent at any time $$t\in[0,T]$$, once the value of the information signal $$\psi$$ is available (ii) the stochastic factor cannot be directly traded, but it is assumed to be also observed at any time $$t\in[0,T]$$. To state differently, the economic agent knows the state of $$(y,m)\in\mathbb{R}\times\mathbb{R}$$ when choosing the optimal control. As a result, we provide the necessary Markovian structure and we are able to apply dynamic programming techniques. Our aim is to maximize the expected utility of the terminal wealth of the informed economic agent, by taking into account the additional information she possesses, that is, we face the problem \begin{equation} u(t,x,y,m) = \sup_{\pi\in\mathcal{A}({\mathbb{G}};T)} \mathbb{E}\Big[U\left(X^{\pi}(T)\right) \big| \psi \Big], \end{equation} (3.1) subject to the dynamic constraint (2.11) and with initial condition $$\left(X^{\pi}(t), Y(t), m(t)\right) = (x,y,m)\in\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}$$. 3.2. Hamilton–Jacobi–Bellman equation of insider’s wealth In order to solve the problem (3.1) and explore the Markovian structure of our model in the enlarged state space representation, given by equation (2.11), we use dynamic programming techniques to derive the associated Hamilton–Jacobi–Bellman equation for the value function. We will employ the following notation. Let $$z=(x,y,m)\in\mathbb{R}_{+}\times\mathbb{R}^{2}$$ and $$t\in[0,T]$$ and consider functions $$u:\mathbb{S}\equiv[0,T]\times\mathbb{R}_{+}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$$ with values $$u(t,z) = u(t,x,y,m)$$. As usual, subscripts with respect to a variable denote partial differentiation of the function with respect to this variable. We will also denote by $$\mathcal{C}^{1,2}$$ the space of functions $$u$$, as above, which are once continuously differentiable with respect to the time variable and twice continuously differentiable with respect to the space variable. The generator for the state process (2.11) subject to the choice $$\pi(t)$$ admits the form \begin{align*} \mathcal{L}^{\pi}h(t,x,y,m) &= \Big[r+ \pi\mu(y) + \pi\sigma(y)m\Big]xh_{x} + \Big[\alpha(y) + \rho\beta(y)m\Big]h_{y}\nonumber\\ &\quad + \frac{1}{2}\Big[ \pi^{2}x^{2}\sigma^{2}(y)h_{xx}+\beta^{2}(y)h_{yy}+f^{2}(t)h_{mm}\Big]\\ &\quad +\pi x \sigma(y)f(t)h_{xm} + \rho\beta(y)f(t)h_{ym} + \pi x \sigma(y)\rho\beta(y)h_{xy}, \nonumber \end{align*} where $$h\in\mathcal{C}^{1,2}$$. Thus, if $$u\in \mathcal{C}^{1,2}$$ one can show that the value function $$u$$ satisfies the following Hamilton–Jacobi–Bellman equation \begin{equation} \begin{aligned} u_{t} + \sup_{\pi\in\mathcal{A}({\mathbb{G}};T)}\mathcal{L}^{\pi}u(t,x,y,m) &= 0\\ u(T,x,y,m) &=U(x). \end{aligned} \end{equation} (3.2) As a result of the state space enlargement we have introduced, we are able to characterize the value function and the optimal investment strategy in terms of the solution of an appropriate Hamilton–Jacobi–Bellman equation. Theorem 3.1 Consider the fully non-linear partial differential equation \begin{equation} \begin{aligned} u_{t}&+rxu_{x}+\alpha(y)u_{y}-\frac{1}{2}\left[\frac{\mu(y)}{\sigma(y)}+m\right]^{2}\frac{u_{x}^{2}}{u_{xx}} +\frac{1}{2}\beta^{2}(y)\left[u_{yy}-\rho^{2}\frac{u_{xy}^{2}}{u_{xx}}\right] +\frac{1}{2}f^{2}(t)\left[u_{mm}-\frac{u_{xm}^{2}}{u_{xx}}\right]\\ &+ \rho\beta(y)m\left[u_{y}-\frac{u_{x}u_{x y}}{u_{xx}}\right]+\rho\beta(y)f(t)\left[u_{ym}-\frac{u_{xm}u_{x y}}{u_{xx}}\right]-\rho\beta(y)\frac{\mu(y)}{\sigma(y)}\frac{u_{x}u_{xy}}{u_{xx}}\\ &-f(t)\left[\frac{\mu(y)}{\sigma(y)}+m\right]\frac{u_{x}u_{xm}}{u_{xx}}=0. \end{aligned} \end{equation} (3.3) If (3.3) admits a solution $$\hat{u}\in\mathcal{C}^{1,2,2,2}(\mathbb{S})$$ and furthermore the function \begin{equation} \pi^{*}(t,x,y,m)=\left\{-\left[\frac{\mu(y)}{\sigma(y)} +m\right]\frac{\hat{u}_{x}}{\hat{u}_{xx}}-\rho\beta(y)\frac{\hat{u}_{xy}}{\hat{u}_{xx}} - f(t)\frac{\hat{u}_{xm}}{\hat{u}_{xx}}\right\}\frac{1}{x\sigma(y)}, \end{equation} (3.4) provides an admissible control law, then $$\hat{u}$$ is the value function for the control problem and $$\pi^{*}$$ is the optimal feedback control law. Proof. For each quadruple $$(t,x,y,m)\in\mathbb{S}$$ we must solve the following equation \begin{eqnarray} && u_{t}+ rxu_{x} + \Big[\alpha(y) + \rho\beta(y)m\Big]u_{y} + \frac{1}{2}\Big[\beta^{2}(y)u_{yy} + f^{2}(t)u_{mm}\Big]+ \rho\beta(y)f(t)u_{ym} \notag\\[-9pt]\\[-3pt] &&\quad+\max_{\pi} \left\{ \Big[\pi\mu(y) + \pi\sigma(y)m\Big]xu_{x}+\frac{1}{2}\pi^{2} x^{2} \sigma^{2}(y)u_{xx} +\pi x\sigma(y)f(t)u_{xm} + \pi x \sigma(y)\rho\beta(y) u_{xy} \right\}=0,\notag \end{eqnarray} (3.5) with boundary condition $$u(T,x,y,m)=U(x)$$. Assume that the maximum in equation (3.5) is attained in the interior of the control region (assuming of course that the control set $$\Pi$$ has a non-empty interior $$\Pi^{0}$$), that is, $$\pi^{*}(t,x,y,m)\in \Pi^{0}$$ for all $$(t,x,y,m)\in\mathbb{S}$$. Differentiating the above expression with respect to $$\pi$$ and setting the derivative equal to zero, gives the candidate optimal control \begin{equation} \hat{\pi}(t,x,y,m)=\left\{-\left[\frac{\mu(y)}{\sigma(y)} +m\right]\frac{u_{x}}{u_{xx}}-\rho\beta(y)\frac{u_{xy}}{u_{xx}} - f(t)\frac{u_{xm}}{u_{xx}}\right\}\frac{1}{x\sigma(y)}. \end{equation} (3.6) In this case, if we place this expression back in equation (3.5), we arrive to the non-linear partial differential equation (3.3). Assume now that the partial differential equation (3.3) admits a classical solution $$\hat{u}\in\mathcal{C}^{1,2,2,2}(\mathbb{S})$$ that satisfies the differentiability conditions $$\hat{u}_{xx},\hat{u}_{yy},\hat{u}_{mm}<0$$ and $$\hat{u}_{x},\hat{u}_{y},\hat{u}_{m}>0$$. By substituting this solution back in equation (3.6) leads to equation (3.4). If moreover $$\hat{\pi}(t,x,y,m)\in \Pi^{0}$$, then $$\pi^{*}(t,x,y,m)$$ coincides with $$\hat{\pi}(t,x,y,m)$$. The rest is a straightforward application of the verification theorem and heavily relies on Itô’s lemma and standard arguments5. For a complete proof, we refer the interested reader to Chapter 3 of Pham (2009) □ Remark 3.2 Equation (3.3) may not in general admit smooth solutions, in which case it can be shown that the value function of the optimal control problem corresponds to a weaker notion of solution (viscosity solution) for which the verification theorem becomes more subtle (see e.g. Crandall et al., 1992 or Fleming & Soner, 2006). However, for a large class of utility functions of interest in applications, smooth solutions of (3.3) can be explicitly constructed (cf. Section 4 as well as the Supplementary Material), thus rendering Theorem 3.1 a useful tool in the solution of the corresponding optimal control problem. Remark 3.3 Consider the maximization problem $$\sup_{\pi\in\Pi}M(\pi)$$, where the function $$\pi\mapsto M(\pi)$$ is defined as \begin{equation} \begin{aligned} M(\pi):=\Big[\pi\mu(y) + \pi\sigma(y)m\Big]x\hat{u}_{x}+\frac{1}{2}\pi^{2} x^{2} \sigma^{2}(y)\hat{u}_{xx} +\pi x\sigma(y)f(t)\hat{u}_{xm} + \pi x \sigma(y)\rho\beta(y) \hat{u}_{xy}. \end{aligned} \end{equation} (3.7) If $$\hat{\pi}(t,x,y,m)\notin \Pi^{0}$$, then $$\pi^{*}(t,x,y,m)$$ is attained at some point at $$\partial \Pi$$, since, under the basis of our assumptions ($$\hat{u}_{xx}<0$$), the function $$M(\pi)$$ is strictly concave on the closed convex set $$\Pi$$. Let us for example consider the simple case $$\Pi=[0,1]$$. This case corresponds to no short-selling the risky asset or borrowing. Additionally, we define the sets \begin{equation} \begin{aligned} \mathfrak{U}_{1}&\equiv \left\{(t,x,y,m)\in\mathbb{S}: \hat{\pi}(t,x,y,m)\geq 1\right\}\\ \mathfrak{U}_{2}&\equiv \left\{(t,x,y,m)\in\mathbb{S}: \hat{\pi}(t,x,y,m)\leq 0\right\}\\ \mathfrak{U}_{3}&\equiv \left\{(t,x,y,m)\in\mathbb{S}: 0<\hat{\pi}(t,x,y,m)<1\right\}\!. \end{aligned} \end{equation} (3.8) Then, the optimal investment strategy is given by \begin{equation} \pi^{*}(t,x,y,m)=\begin{cases}1 &\,\, ;(t,x,y,m)\in\mathfrak{U}_{1}\\ \hat{\pi}(t,x,y,m) &\,\, ;(t,x,y,m)\in\mathfrak{U}_{3}\\ 0 &\,\, ;(t,x,y,m)\in\mathfrak{U}_{2},\end{cases} \end{equation} (3.9) and one should be able to check6 that $$\pi^{*}(t,x,y,m)\in\mathcal{A}({\mathbb{G}};T)$$. Remark 3.4 From equation (3.4) and by taking into account equations (2.9) and (2.10) we note that the optimal investment strategy depends on the information drift $$m$$ and the stochastic factor $$y$$. This is something very natural since: (i) we have assumed that the agent has an a priori knowledge of the value of the random variable $$\psi$$ and $$m$$ is the information drift induced by this future information. This of course means that the agent as an informed trader is taking advantage of her additional knowledge, as expected (ii) we have assumed that the stock prices are influenced by the external factor $$y$$. Hence, as also expected, the optimal investment strategy should take into account this dependence. Remark 3.5 From equation (3.4) we can see that the optimal investment policy consists of three terms, namely: \begin{align*} \pi^{*;0}(t,x,y) &= -\frac{\mu(y)}{x\sigma^{2}(y)}\frac{\hat{u}_{x}}{\hat{u}_{xx}}\\ \pi^{*;1}(t,x,y,m) &=\left[-m\frac{\hat{u}_{x}}{\hat{u}_{xx}} -f(t)\frac{\hat{u}_{xm}}{\hat{u}_{xx}}\right]\frac{1}{x\sigma(y)}\\ \pi^{*;2}(t,x,y) &=-\rho\frac{\beta(y)}{x\sigma(y)}\frac{\hat{u}_{xy}}{\hat{u}_{xx}}. \end{align*} The first component, $$\pi^{*;0}(t,x,y)$$, is known as the Merton investment strategy. This myopic behaviour corresponds to the investment policy followed by the economic agent under symmetric information in markets in which the investment opportunity set remains constant through time.7 Under the basis of our assumptions ($$x, \hat{u}_{x}>0$$ and $$\hat{u}_{xx}<0$$), the Merton portfolio is always positive for a non-zero market price of risk ($$\mu(y)>0$$). The second component, $$\pi^{*;1}(t,x,y,m)$$, is called the information signal hedging demand. It represents the additional investment caused by the knowledge of the information signal $$\psi$$. To be more precise, $$\pi^{*;1}(t,x,y,m)$$ behaves like a correction term to the Merton portfolio $$\pi^{*;0}(t,x,y)$$ so as to take into account the effect of the additional information. This term does not have a constant sign, for the signs of the mixed derivative $$\hat{u}_{xm}$$ and the information variable $$m$$ are not definite. Note that, if the information signal observed is completely imprecise, that is, it is only consisted of noise, this term should vanish (cf. Remark 4.4). The third component, $$\pi^{*;2}(t,x,y)$$, is called the excess hedging demand. It represents the additional investment caused by the presence of the stochastic factor. This term also does not have a constant sign, for the signs of the mixed derivative $$\hat{u}_{xy}$$ and the correlation coefficient $$\rho$$ are not definite. Note that the excess hedging demand vanishes in the uncorrelated case ($$\rho=0$$) and also when the volatility of the stochastic factor process is zero ($$\beta(y)=0$$), something however that is not allowed in our case. As already mentioned above (cf. Remark 4.4), when the inside information is not taken into account, the component $$\pi^{*;1}(t,x,y,m)$$ vanishes. In this case, the private information does not have anything to offer hence it is ignored; the manager operates under the filtration $$\mathbb{F}$$ and the optimal investment strategy is given by \begin{equation*} \pi^{*}(t,x,y) = \pi^{*;0}(t,x,y) + \pi^{*;2}(t,x,y). \end{equation*} 4. The case of the exponential utility function In the present section, in order to demonstrate the effectiveness of the proposed approach, we provide solutions in feedback form for the optimal investment strategy and the optimal value function, for the special case of the exponential utility function. In other words, we assume that the informed economic agent has exponential preferences, that is, a utility function of the form \begin{equation} U(x) = -\frac{1}{\gamma}e^{-\gamma x}, \end{equation} (4.1) where $$\gamma>0$$. This function has a constant absolute risk aversion parameter $$\gamma$$8 and plays an important role in both finance and insurance. Theorem 4.1 Assume exponential preferences (equation (4.1)). Then, the value function for the informed economic agent which is also the solution of the stochastic optimal control problem (3.1), admits the form \begin{equation} u(t,x,y,m)=-\frac{1}{\gamma}\exp\left[-\gamma xe^{r(T-t)} + g(t,y,m)\right], \end{equation} (4.2) where the function $$g$$ satisfies the following non-linear partial differential equation \begin{equation} \begin{aligned} g_{t} &+ \left[\alpha(y) - \rho\beta(y)\frac{\mu(y)}{\sigma(y)}\right]g_{y} - \frac{1}{2}\left[\frac{\mu(y)}{\sigma(y)} + m\right]^{2} +\frac{1}{2}\beta^{2}(y)\Big[(1-\rho^{2})g_{y}^{2}+g_{yy}\Big] \\ &-f(t)\left[\frac{\mu(y)}{\sigma(y)} + m\right]g_{m}+ \rho\beta(y) f(t)g_{ym}+\frac{1}{2}f^{2}(t)g_{mm} =0. \end{aligned} \end{equation} (4.3) with boundary condition $$g(T,y,m)=0$$. In this case, the optimal investment strategy is given in feedback form by \begin{equation} \pi^{*}(t,x,y,m)=\left[\left(\frac{\mu(y)}{\sigma(y)} + m\right) + \rho\beta(y)g_{y}+f(t) g_{m}\right]\frac{e^{-r(T-t)}}{\gamma x\sigma(y)}. \end{equation} (4.4) Proof. Suppose the partial differential equation (3.3) admits a classical solution $$u\in \mathcal{C}^{1,2,2,2}(\mathbb{S})$$ for every quadruple $$(t,x,y,m)\in\mathbb{S}$$. We look for solutions using the following ansatz: \begin{equation*} u(t,x,y,m) = -\frac{1}{\gamma}\exp\left\{ -\gamma x e^{r(T-t)} +g(t,y,m)\right\}\!, \end{equation*} where $$g(t,y,m)$$ is a suitable function with boundary conditions $$g(T,y,m)=0$$ (this follows from the boundary condition $$u(T,x,y,m)=U(x)$$), which will be determined later. Differentiating this trial solution with respect to $$(t,x,y,m)$$, yields \[ \begin{array}{ll} u_{t}(t,x,y,m)=u(t,x,y,m)\left[r\gamma xe^{r(T-t)} + g^{'}(t) \right] & u_{xm}(t,x,y,m)=u(t,x,y,m)\left[-\gamma e^{r(T-t)}\right]g_{m}\\ u_{x}(t,x,y,m)=u(t,x,y,m)\left[-\gamma e^{r(T-t)}\right] & u_{xy}(t,x,y,m)=u(t,x,y,m)\left[-\gamma e^{r(T-t)}\right]g_{y} \\ u_{m}(t,x,y,m) =u(t,x,y,m)g_{m} & u_{mm}(t,x,y,m)=u(t,x,y,m)\Big[g_{m}^{2} + g_{mm}\Big]\\ u_{y}(t,x,y,m) =u(t,x,y,m)g_{y} & u_{yy}(t,x,y,m)=u(t,x,y,m)\Big[g_{y}^{2} + g_{yy}\Big]\\ u_{xx}(t,x,y,m)=u(t,x,y,m)\left[\gamma e^{r(T-t)}\right]^{2} & u_{ym}(t,x,y,m)=u(t,x,y,m)\Big[g_{y}g_{m} + g_{ym}\Big]. \end{array}\] Substituting the above expressions in equation (3.3) leads to the non-linear partial differential equation (4.3) and in equation (3.4) gives the optimal control law (4.4) □ Corollary 4.1 In the complete market case $$(\rho=1)$$, the non-linear partial differential equation (4.3), reduces to the linear parabolic partial differential equation \begin{equation} \begin{aligned} g_{t} &+ \left[\alpha(y) -\beta(y)\frac{\mu(y)}{\sigma(y)}\right]g_{y} - \frac{1}{2}\left[\frac{\mu(y)}{\sigma(y)} + m\right]^{2} +\frac{1}{2}\beta^{2}(y)g_{yy}-f(t)\left[\frac{\mu(y)}{\sigma(y)} + m\right]g_{m}\\ &+ \beta(y) f(t)g_{ym}+ \frac{1}{2}f^{2}(t)g_{mm}=0, \end{aligned} \end{equation} (4.5) and in this case, the optimal investment decision is given by \begin{equation} \pi^{*}(t,x,y,m)=\left[\left(\frac{\mu(y)}{\sigma(y)} + m\right) + \beta(y)g_{y}+f(t) g_{m}\right]\frac{e^{-r(T-t)}}{\gamma x\sigma(y)}. \end{equation} (4.6) 4.1. A fully worked example In Theorem 4.1, we characterized the value function of the problem (3.1) as the solution of a non-linear partial differential equation and moreover derived the optimal investment policy, in the case of the exponential utility function (4.1). However, no further conclusions can be extracted since the form of the functions ($$\mu,\alpha,\beta$$ and $$\sigma$$) of the underlying driving system and the information signal $$\psi$$ are not explicitly defined. In the present section, for the sake of a more comprehensive analysis, we assume a specific form for the above parameters and present a fully worked example. To be more precise, in order to keep things as simple as possible, we adopt the complete market framework $$(\rho=1)$$, and assume that the parameters for the risky asset (2.2) and the stochastic factor process (2.3), are defined as \begin{align*} \mu(y) &= \mu_{0}\left(|y|+\delta\right)\\ \sigma(y) &= \sigma_{0}\left(|y|+\delta\right)\\ \alpha(y) &= \alpha_{0}(\theta-y)\\ \beta(y) &= \beta_{0}, \end{align*} where $$\mu_{0},\sigma_{0},\alpha_{0},\beta_{0}>0$$ and $$\theta\in\mathbb{R}$$. The parameter $$\delta$$ is a very small constant needed in order for Assumption 2.1 to be satisfied. In other words, the dynamics of the risky asset evolve according to the stochastic differential equation \begin{equation} \begin{aligned} \frac{dS(t)}{S(t)}&= \Big[r + \mu_{0}\Big(|Y(t)|+\delta\Big)\Big]dt + \sigma_{0}\Big(|Y(t)|+\delta\Big) dW(t)\\ S(0)&=S_{0}>0, \end{aligned} \end{equation} (4.7) while the stochastic factor process is driven by the Ornstein–Uhlenbeck process \begin{equation} \begin{aligned} dY(t)&= \alpha_{0}\left(\theta-Y(t)\right)dt + \beta_{0} dW(t)\\ Y(0)&=Y_{0} \in\mathbb{R}. \end{aligned} \end{equation} (4.8) The choice for the functions $$\mu,\sigma,\alpha$$ and $$\beta$$ for this example was made so that a realistic model which is analytically tractable within the reasonable page limits of this publication can be presented. Of course, more general options for the functions $$\mu,\sigma,\alpha$$ and $$\beta$$ can be used, allowing for more complex models and scenarios, that can be accommodated within Theorem 3.1. These are left to the interested reader. In what follows in order to avoid cluttering of notation we will use the notation $$\mu$$, $$\sigma$$, $$\alpha$$, $$\beta$$ for the constants $$\mu_0$$, $$\sigma_0$$, $$\alpha_0$$ and $$\beta_0$$. Remark 4.1 The above specific choice for the market parameters is mainly driven by the requirements of Assumption 2.1 together with the fact that an Ornstein–Uhlenbeck type process is a very popular choice for the stochastic factor (e.g. interest rate or stochastic volatility), see for example Bielecki & Pliska (2001), Delong & Klüppelberg (2008) or Fouque et al. (2000) and references therein. However, note that the requirements of Assumption 2.1, especially the globally Lipschitz assumption, exclude some well-known models such as the Heston or the Hull and White stochastic volatility models. At this point, take into account that the model described by equations (4.7) and (4.8), is not aimed to serve as the most realistic model (if there exists one) but to give an insight on how the proposed framework works in practice, keeping of course in line with Assumption 2.1. This model will not only provide explicit closed form solutions for the value function and the optimal investment strategy but also lay the groundwork for a detailed numerical analysis in Section 5. Additionally, we consider $$T_{0}=T$$ within the setting described in Section 2.3, and assume that the information signal $$\psi$$ admits the form \begin{equation} \psi = \lambda W(T) + (1-\lambda)\epsilon, \end{equation} (4.9) where $$\epsilon$$ is a standard normal random variable, independent of $$W$$. In this case, the manager possesses information about the final value of the Brownian motion $$W(t)$$ in the trading interval $$[0,T]$$, which is distorted by some observation noise $$\epsilon$$. A value of $$\lambda=1$$ means that the manager explicitly knows the value of $$W(T)$$, whereas, a value of $$\lambda=0$$ means that the manager observes nothing but noise. In this setting, we are only interested in non-trivial case $$0\leq\lambda<1$$, that is, the available information comes with noise. For the case at hand, it is well known (see e.g. Fei & Wu, 2003 or Pikovsky & Karatzas, 1996), that the information drift admits the form \begin{equation} m(t) = \frac{\big[\psi - \lambda W(t)\big]\lambda}{\lambda^{2}(T-t) + (1-\lambda)^{2}}, \end{equation} (4.10) for every $$t\in[0,T]$$ and $$0\leq\lambda<1$$. In fact, equation (4.10) is fully compatible with our Assumptions regarding the information drift. To be more precise, concerning Assumption 2.2, we have the following well-known result. Remark 4.2 (Pikovsky & Karatzas, 1996) The information drift (4.10), satisfies \begin{equation*} \mathbb{E}\left[\int_{0}^{T}m^{2}(s)ds\right]\leq\frac{C}{(1-\lambda)^{2}}<\infty, \end{equation*} for $$\lambda<1$$ and some constant $$C>0$$. Remark 4.3 In the adopted framework, the portfolio manager observes, from the beginning of the trading interval, a signal $$\psi$$ concerning the terminal value of the underlying Brownian motion $$W$$. However, this signal is not clear, in the sense that it is subjected to some observation noise $$\epsilon$$. As a result, for any value of $$\lambda\in[0,1)$$, the final value $$W(T)$$ is not precisely known. This fact prevents us from resorting to the classical theory of Brownian bridge enlargements. Concerning the justification of Assumption 2.3, we have the next result. Proposition 4.1 For the information drift defined by equation (4.10), we have that \begin{align} f(t) = -\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}} \end{align} (4.11) and \begin{align} \hspace{-2.8em}m(0)= \frac{\lambda \psi}{\lambda^{2}T + (1-\lambda)^{2}}. \end{align} (4.12) Proof. First, note that according to equation (2.9), under the enlarged observed filtration $$\mathbb{G}$$, the process $$W(t)$$ is an Itô process with decomposition given by \begin{equation*} dW(t) = d\widetilde{W}(t) + m(t)dt, \end{equation*} where $$\widetilde{W}(t)$$ is a $$\mathbb{G}$$-Brownian motion. A straightforward application of Itô’s lemma to the $$\mathcal{C}^{1,2}\left([0,T]\times\mathbb{R}\right)$$ function $$m(t,W(t))$$, yields (observe equation (4.10)) \begin{align*} dm(t,W(t)) &= \frac{\partial m(t,W(t))}{\partial t}dt + \frac{\partial m(t,x)}{\partial x}\Big|_{x=W(t)}dW(t) + \frac{1}{2}\frac{\partial^{2} m(t,x)}{\partial x^{2}}\Big|_{x=W(t)}(dW(t))^{2} \nonumber \\ &= \frac{\lambda^{3}\left[\psi - \lambda W(t)\right]}{\left[\lambda^{2}(T-t) + (1-\lambda)^{2}\right]^{2}}dt -\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\Bigg[d\widetilde{W}(t) + \frac{\big[\psi - \lambda W(t)\big]\lambda}{\lambda^{2}(T-t) + (1-\lambda)^{2}}dt \Bigg] \nonumber \\ &=-\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}d\widetilde{W}(t), \end{align*} and equation (4.11) follows by comparing the above equation, with equation (2.10). The proof is concluded by setting in equation (4.10), $$t=0$$ and taking into account that $$W$$ is assumed to be a standard Brownian motion ($$W(0)=0$$). □ Theorem 4.2 Assume exponential preferences (equation (4.1)) and moreover that the risky asset evolves according to the stochastic differential equations (4.7) and (4.8) and that the information signal is given by equation (4.9). The value function for the informed economic agent which is also the solution of the stochastic optimal control problem (3.1), admits the form \begin{equation} u(t,x,y,m)=-\frac{1}{\gamma}\exp\left[-\gamma xe^{r(T-t)} + g(t,y,m)\right], \end{equation} (4.13) where \begin{equation} g(t,y,m) = A_{1}(t)m^{2} + A_{2}(t)m + A_{3}(t)y^{2} + A_{4}(t)y+ A_{5}(t)my + A_{6}(t), \end{equation} (4.14) with \begin{align} A_{1}(t) &= -\frac{T-t}{2(1-\lambda)^{2}}\Big[\lambda^{2}(T-t) + (1-\lambda)^{2}\Big]\\ \end{align} (4.15) \begin{align} A_{2}(t) &= -\frac{\mu}{\sigma}\frac{T-t}{(1-\lambda)^{2}}\Big[\lambda^{2}(T-t) + (1-\lambda)^{2}\Big]\\ \end{align} (4.16) \begin{align} A_{3}(t) &= A_{4}(t)= A_{5}(t) = 0\\ \end{align} (4.17) \begin{align} A_{6}(t) &= \frac{1}{2(1-\lambda)^{2}}\left\{-\lambda^{2}(T-t)+ (1-\lambda)^{2}\log\left[\frac{(1-\lambda)^{2} + \lambda^{2}(T-t)}{(1-\lambda)^{2}}\right]\right\} \\ &\quad-\frac{1}{2}\left(\frac{\mu}{\sigma}\right)^{2}(T-t) \left[1 + \frac{\lambda^{2}}{(1-\lambda)^{2}}(T-t)\right].\nonumber \end{align} (4.18) In this case, the optimal investment strategy for the informed economic agent is given by the feedback rule \begin{equation} \pi^{*}(t,x,y,m)=\frac{\mu + \sigma m}{\sigma}\frac{\lambda^{2}(T-t) + (1-\lambda)^{2}}{(1-\lambda)^{2}}\frac{e^{-r(T-t)}}{\gamma x\sigma(|y|+\delta)}. \end{equation} (4.19) Proof. For the case at hand (note that $$f(t)$$ is given by equation (4.11)), equation (4.5) leads to \begin{equation} \begin{aligned} g_{t} &+ \left[\alpha(\theta-y) - \beta\frac{\mu}{\sigma}\right]g_{y} - \frac{1}{2}\left(\frac{\mu}{\sigma} + m\right)^{2} +\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\left(\frac{\mu}{\sigma}+m\right)g_{m}\\ &+\frac{1}{2}\left[\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\right]^{2}g_{mm}-\frac{\beta\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}g_{ym} +\frac{1}{2}\beta^{2}g_{yy} =0. \end{aligned} \end{equation} (4.20) We conjecture a solution to the linear parabolic partial differential equation (4.20), with the following form \begin{equation*} g(t,y,m) = A_{1}(t)m^{2} + A_{2}(t)m + A_{3}(t)y^{2} + A_{4}(t)y+ A_{5}(t)my + A_{6}(t), \end{equation*} where $$A_{1}(t),A_{2}(t),A_{3}(t),A_{4}(t),A_{5}(t)$$ and $$A_{6}(t)$$ are suitable functions to be determined later with boundary condition $$A_{1}(T)=A_{2}(T)=A_{3}(T)=A_{4}(T)=A_{5}(T)=A_{6}(T)=0$$ (this follows from the boundary condition $$g(T,y,m)=0$$). Substituting this trial solution in equation (4.20) we get the following ordinary differential equations \begin{align} &A_{1}'(t) + \frac{2\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}A_{1}(t) - \frac{1}{2}=0 \\ \end{align} (4.21a) \begin{align} &A_{2}'(t) + \frac{A_{2}(t)\lambda^{2}}{\lambda^{2}(T-t)+(1-\lambda)^{2}} + \frac{2\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\frac{\mu}{\sigma}A_{1}(t)+\left(\alpha\theta-\beta\frac{\mu}{\sigma}\right)A_{5}(t) - \frac{\mu}{\sigma}=0 \\ \end{align} (4.21b) \begin{align} &A_{3}'(t) - 2\alpha A_{3}(t)=0 \\ \end{align} (4.21c) \begin{align} &A_{4}'(t) - \alpha A_{4}(t) + 2\left(\alpha\theta - \beta\frac{\mu}{\sigma}\right)A_{3}(t) + \frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\frac{\mu}{\sigma}A_{5}(t) =0 \\ \end{align} (4.21d) \begin{align} & A_{5}'(t) - \Bigg[a - \frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\Bigg]A_{5}(t) = 0 \\ \end{align} (4.21e) \begin{align} & A_{6}'(t) + \left(\alpha\theta - \beta\frac{\mu}{\sigma}\right)A_{4}(t) -\frac{1}{2}\left(\frac{\mu}{\sigma}\right)^{2} +\beta^{2}A_{3}(t) + \left[\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\right]^{2}A_{1}(t) \nonumber \\ &\qquad +\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\frac{\mu}{\sigma}A_{2}(t) - \frac{\beta\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}A_{5}(t) =0. \end{align} (4.21f) In order to solve the above ordinary differential equations, we proceed as follows: (i) we solve equations (4.21a), (4.21c) and (4.21e), (ii) we substitute the solutions we found for equations (4.21a) and (4.21e) into equation (4.21b) and then solve it, (iii) we substitute the solutions we found for equations (4.21c) and (4.21e) into equation (4.21d) and then solve it (iv) and finally, substitute the solutions for equations (4.21a–4.21e) into equation (4.21f) and then solve it. This eventually yields to equation (4.13) which is the optimal value function associated with the stochastic optimal control problem at hand. Regarding the optimal investment strategy, from equation (4.6), adapted to our specific example, we observe that \begin{align*} &\pi^{*}(t,x,y,m)\\ &\quad=\left[\left(\frac{\mu(y)}{\sigma(y)} + m\right) + \beta(y)g_{y}+f(t) g_{m}\right]\frac{e^{-r(T-t)}}{\gamma x\sigma(y)} \\ &\quad=\left[\frac{\mu + \sigma m}{\sigma} + \beta\left(2A_{3}(t)y + A_{4}(t) + A_{5}(t)m\right) - \frac{\lambda^{2}\left(2A_{1}(t)m + A_{2}(t) + A_{5}(t)y\right)}{\lambda^{2}(T-t)+(1-\lambda)^{2}}\right]\frac{e^{-r(T-t)}}{\gamma x\sigma(|y|+\delta)} \\ &\quad=\left[\frac{\mu + \sigma m}{\sigma} - \frac{\lambda^{2}}{\lambda^{2}(T-t)+(1-\lambda)^{2}}\left(2A_{1}(t)m + A_{2}(t)\right)\right]\frac{e^{-r(T-t)}}{\gamma x\sigma(|y|+\delta)} \\ &\quad=\frac{\mu + \sigma m}{\sigma}\frac{\lambda^{2}(T-t) + (1-\lambda)^{2}}{(1-\lambda)^{2}}\frac{e^{-r(T-t)}}{\gamma x\sigma(|y|+\delta)}, \end{align*} which follows from equation (4.13) and the solution of the ordinary differential equations (4.21a)–(4.21f). This completes the proof □ Remark 4.4 Regarding the optimal investment strategy, by observing equation (4.19), we distinguish the following two interesting cases concerning the possible values of $$\lambda$$: $$\underline{\text{When}\,{\lambda=0}}$$, the information signal becomes $$\psi=\epsilon$$, that is, it is only consisted of noise. In this case, the optimal investment strategy is given in feedback form by \begin{equation*} \pi^{*}(t,x,y) =\frac{\mu}{\sigma}\frac{e^{-r(T-t)}}{\gamma x\sigma(|y|+\delta)}, \end{equation*} (and is obtained by setting in equations (4.10) and (4.19), $$\lambda=0$$). This is the optimal Markovian control law associated with the stochastic optimal control problem \begin{equation*} u(t,x,y) = \sup_{\pi\in\mathcal{A}({\mathbb{F}};T)} \mathbb{E}\Big[U\left(X^{\pi}(T)\right) \big| \mathscr{F}_{t}\Big], \end{equation*} subject to the state process \begin{equation*} \begin{aligned} \frac{dX^{\pi}(t)}{X^{\pi}(t)} &= \Big[r + \pi(t)\mu\left(|Y(t)|+\delta\right)\Big]dt + \pi(t)\sigma\left(|Y(t)|+\delta\right)dW(t) \\ dY(t) &=\alpha\left(\theta-Y(t)\right)dt + \beta dW(t), \end{aligned} \end{equation*} since in this case $$m(t)=0$$ for every $$t\in[0,T]$$. Stated differently, when $$\lambda=0$$ the economic agent ignores the information signal, as it has nothing special to offer, and operates under the publicly available information flow $$\mathbb{F}$$. $$\underline{\text{When}\,{\lambda=1}}$$, the information signal becomes $$\psi=W(T)$$, that is, the final value of the underlying Brownian motion driving uncertainty in the stock price process (4.7), is explicitly available at the agent’s disposal from the very beginning of the trading interval. In this case, \begin{equation} m(t) \longrightarrow \frac{W(T)-W(t)}{T-t}. \end{equation} (4.22) As $$t\rightarrow T^{-}$$ the right hand side in equation (4.22) behaves like the non-existing derivative of the Brownian motion $$\left\{W(t),\, t\geq 0\right\}$$ at the point $$t=T$$, hence the aforementioned strategy is impossible. Such a strategy would be characterized by wild fluctuations around $$T$$ and this behaviour might act as an indicator the economic agent operates under the influence of private information. Moreover, it is well known from Pikovsky & Karatzas (1996), that in this case the value of the problem becomes infinite and arbitrage opportunities arise. For more information on this subject, we refer the interested reader to the article of Liu & Longstaff (2004). Remark 4.5 If we set in equation (4.9), $$\lambda=1$$ and consider the future time $$T_{0}>T$$, we retrieve the case $$\psi = W(T_{0})$$, where $$T_{0}>T$$. That is, the manager explicitly knows the value of the underlying Brownian motion $$W(t)$$ at the future time $$T_{0}$$. In this case, it is well-known (see e.g. Pikovsky & Karatzas, 1996; Mansuy & Yor, 2006) that the information drift is given by \begin{equation} m(t) = \frac{\psi - W(t)}{T_{0}-t}. \end{equation} (4.23) Furthermore, Assumption 2.2 is satisfied for any $$T<T_{0}$$, and by an application of Itô’s lemma to the $$\mathcal{C}^{1,2}\left([0,T]\times\mathbb{R}\right)$$ function $$m(t,W(t))$$, defined in equation (4.23), we get that \begin{equation} f(t) = -\frac{1}{T_{0}-t} \end{equation} (4.24) and \begin{equation} m(0) = \frac{\psi}{T_{0}}. \end{equation} (4.25) As a result, we can see that Assumption 2.3 is again justified. In order to have a complete picture for the problem at hand, we provide a result similar to Theorem 4.2, adapted to the setting of Remark 4.5. Theorem 4.3 Assume exponential preferences (equation (4.1)) and moreover that the risky asset evolves according to the stochastic differential equations (4.7) and (4.8) and that the information signal is given $$\psi=W(T_{0}), T_{0}>T$$. The value function for the informed economic agent which is also the solution of the stochastic optimal control problem (3.1), admits the form \begin{equation} u(t,x,y,m)=-\frac{1}{\gamma}\exp\left[-\gamma xe^{r(T-t)} + \zeta(t,y,m)\right], \end{equation} (4.26) where \begin{equation} \zeta(t,y,m) = F_{1}(t)m^{2} + F_{2}(t)m + F_{3}(t)y^{2} + F_{4} (t)y+ F_{5}(t)my + F_{6}(t), \end{equation} (4.27) with \begin{align} F_{1}(t)&=-\frac{(T_{0}-t)(T-t)}{2(T_{0}-T)} \\[3pt] \end{align} (4.28) \begin{align} F_{2}(t)&=-\frac{\mu(T_{0}-t)(T-t)}{\sigma(T_{0}-T)} \\[3pt] \end{align} (4.29) \begin{align} F_{3}(t)&=F_{4}(t)=F_{5}(t)=0\\[3pt] \end{align} (4.30) \begin{align} F_{6}(t)&=-\left(\frac{\mu}{\sigma}\right)^{2}\frac{(T_{0}-t)(T-t)}{2(T_{0}-T)} + \frac{1}{2}\Bigg[ \log\left[\frac{T_{0}-t}{T_{0}-T}\right] -\frac{T-t}{T_{0}-T} \Bigg]. \end{align} (4.31) In this case, the optimal investment strategy for the informed agent is given in feedback form by \begin{equation} \pi^{*}(t,x,y,m) = \frac{\mu + \sigma m}{\sigma}\frac{T_{0}-t}{T_{0}-T}\frac{e^{-r(T-t)}}{\gamma x \sigma (|y|+\delta)}. \end{equation} (4.32) Proof. For the case at hand (note that $$f(t)$$ is given by (4.24)), equation (4.5) leads to \begin{equation} \begin{aligned} \zeta_{t} &+ \left[\alpha(\theta-y) - \beta\frac{\mu}{\sigma}\right]\zeta_{y} - \frac{1}{2}\left(\frac{\mu}{\sigma} + m\right)^{2} + \frac{1}{T_{0}-t}\left(\frac{\mu}{\sigma}+m\right)\zeta_{m}\\ &+\frac{1}{2}\left(\frac{1}{T_{0}-t}\right)^{2}\zeta_{mm}-\frac{\beta}{T_{0}-t}\zeta_{ym} +\frac{1}{2}\beta^{2}\zeta_{yy} =0. \end{aligned} \end{equation} (4.33) We conjecture a solution to the linear parabolic partial differential equation (4.33), with the following form \begin{equation*} \zeta(t,y,m) = F_{1}(t)m^{2} + F_{2}(t)m + F_{3}(t)y^{2} + F_{4}(t)y+ F_{5}(t)my + F_{6}(t), \end{equation*} where $$F_{1}(t),F_{2}(t),F_{3}(t),F_{4}(t),F_{5}(t)$$ and $$F_{6}(t)$$ are suitable functions to be determined later with boundary condition $$F_{1}(T)=F_{2}(T)=F_{3}(T)=F_{4}(T)=F_{5}(T)=F_{6}(T)=0$$ (this follows from the boundary condition $$\zeta(T,y,m)=0$$). Substituting this trial solution in equation (4.33), we get the following ordinary differential equations \begin{align} &F_{1}'(t) + \frac{2F_{1}(t)}{T_{0}-t} - \frac{1}{2}=0 \\ \end{align} (4.34a) \begin{align} &F_{2}'(t) + \frac{F_{2}(t)}{T_{0}-t} + \frac{2}{T_{0}-t}\frac{\mu}{\sigma}F_{1}(t)+\left(\alpha\theta-\beta\frac{\mu}{\sigma}\right)F_{5}(t) - \frac{\mu}{\sigma}=0 \\ \end{align} (4.34b) \begin{align} &F_{3}'(t) - 2\alpha F_{3}(t)=0 \\ \end{align} (4.34c) \begin{align} &F_{4}'(t) - \alpha F_{4}(t) + 2\left(\alpha\theta - \beta\frac{\mu}{\sigma}\right)F_{3}(t) + \frac{1}{T_{0}-t}\frac{\mu}{\sigma}F_{5}(t) =0 \\ \end{align} (4.34d) \begin{align} & F_{5}'(t) - \Bigg[a - \frac{1}{T_{0}-t}\Bigg]F_{5}(t) = 0 \\ \end{align} (4.34e) \begin{align} & F_{6}'(t) + \left(\alpha\theta - \beta\frac{\mu}{\sigma}\right)F_{4}(t) -\frac{1}{2}\left(\frac{\mu}{\sigma}\right)^{2} +\beta^{2}F_{3}(t) + \left[\frac{1}{T_{0}-t}\right]^{2}F_{1}(t) \nonumber \\ &\qquad +\frac{1}{T_{0}-t}\frac{\mu}{\sigma}F_{2}(t) - \frac{\beta}{T_{0}-t}F_{5}(t) =0. \end{align} (4.34f) Solving the above system of ordinary differential equations (by following the same approach as in the proof of Theorem 4.2), leads to the optimal value function (4.26) and to the optimal investment strategy (4.32) □ 5. Numerical study of the optimal investment strategy According to the concrete example of Section 4.1, the portfolio manager, from the very beginning of the trading interval, has exclusive access to some additional information which stems from the observation of a private signal $$\psi$$ concerning the value of $$W(T)$$, as it is defined in equation (4.9). However, this signal is not precise, in the sense that it is subject to some normally distributed observation noise $$\epsilon$$. Having the value of $$\psi$$ in mind, the investor, tries to make the best possible estimation about the value of $$Y(T)$$ and in advance about the value of $$S(T)$$. Based on this prediction, she then decides the optimal investment policy to be followed, which in the special case: the financial market is driven by the system of stochastic differential equations (4.7) and (4.8) the information signal admits the form (4.9) and under exponential preferences (equation (4.1)) is given in feedback form by \begin{equation} \pi^{*}(t,X^{*}(t),Y(t),m(t))=\frac{\mu + \sigma m(t)}{\sigma}\frac{\lambda^{2}(T-t) + (1-\lambda)^{2}}{(1-\lambda)^{2}}\frac{e^{-r(T-t)}}{\gamma X^{*}(t)\sigma\left(|Y(t)|+\delta\right)}, \end{equation} (5.1) where $$X^{*}(t)=X^{\pi^{*}}(t)$$ denotes the wealth process under the optimal investment strategy (the dynamics of $$X^{*}(t)$$ can be easily obtained by substituting equation (5.1) in the SDE that describes the wealth of the economic agent, see equation (2.11)). Hence, as far as the simulation of the optimal investment strategy is concerned, special treatment is needed in order to effectively capture the behaviour of the informed agent, with the major obstacle being the simulation of $$X^{*}(t), Y(t)$$ and $$m(t)$$. In this endeavour, we follow the next steps: (i) Numerically solve the stochastic differential equations for $$X^{*}(t), Y(t)$$ and $$m(t)$$ by employing an Euler–Maruyama scheme. In order to implement the method, for a time step of size $$\Delta t=T/N$$ with $$N=2^{11}$$ points, we define the step size in the Euler–Maruyama scheme as $$\delta t=\Delta t$$. (ii) Based on step (i), calculate the mean over a large number of realizations of the optimal investment strategy $$\pi^{*}$$, as defined in equation (5.1). An appropriate method to attack this problem is the Monte Carlo simulation. In order to implement the Monte Carlo method, we simulate a large number M of paths of $$\pi^{*}$$ in the time interval $$[0,T]$$ and at each time point we plot the average of M different values. We also use for each path $$N = 2^{\alpha}$$ number of points (here $$N=2^{11}$$ and $$M=6000$$ paths). In what follows, unless stated otherwise, we let $$\Pi=[0,1]$$, $$T=1$$ year, $$X(0)=1.5$$ and $$\gamma=1.5$$. The parameters of the financial market are chosen as $$\mu=10\%$$, $$r=6\%$$, $$\sigma=40\%$$ and $$\delta=10^{-3}$$. The parameters for the external factor are chosen as $$Y(0)=1.5$$, $$\alpha=2$$, $$\theta=1$$ and $$\beta=40\%$$. Remark 5.1 When employing an Euler–Maruyama scheme, one is usually interested in the convergence of the scheme, that is, if the simulated solution converges to the exact solution of the stochastic differential equation at hand. Typically, the parameters must satisfy some globally Lipschitz and a non-explosion condition (see e.g. Gilsing & Shardlow, 2007). In our case, convergence seems to hold, at least in a weak sense. However, we do not examine this matter any further, since it exceeds the scope of the current work. 5.1. Simulation results From Fig. 1, we have some very interesting findings: Fig. 1. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths. In this case we consider $$\lambda=0.2$$. (b) Average path of 6000 optimal investment strategy paths for various values of the parameter $$\lambda$$. In this case we consider $$\psi=0.2$$. Fig. 1. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths. In this case we consider $$\lambda=0.2$$. (b) Average path of 6000 optimal investment strategy paths for various values of the parameter $$\lambda$$. In this case we consider $$\psi=0.2$$. From Fig. 1(a), we can see that as the level of the value of $$\psi$$ increases, the economic agent is expected to invest more in the stock market compared to the symmetric information case. This seems quite natural, since an increasing level for the information signal leads to an increasing value for $$W(T)$$ which in advance leads to an optimistic prediction for $$S(T)$$. With this prediction at hand, the economic agent expects higher returns for the underlying stock leading to more interest in the risky market. On the other hand, a decreasing value for $$\psi$$ leads to pessimistic predictions for $$S(T)$$ and as a result the economic agent turns her attention to the risk-free asset in an attempt to seek safer investment opportunities. As the parameter $$\lambda$$ increases, the investor is expected to follow what the private information dictates. That is, she is expected to invest more in the risky asset (as the level of the information signal increases) and to turn her attention in the bond market (as the level of the information signal decreases). On the other hand, as $$\lambda\rightarrow 0$$, as already stated in Remark 4.4, the information signal becomes very imprecise and has nothing useful to offer. In this case, from Fig. 1(b), we can see that the optimal investment strategy of the informed agent converges to the behaviour of the honestly informed agent. In fact, this result coincides with the findings of Remark 4.4. From Fig. 2(a,b), we have the following interesting results: From Fig. 2(a), we can see that as the risk aversion parameter $$\gamma$$ increases, the economic agent is expected to invest more in the risk-free asset. This also seems very natural, since in this case the agent is too risk averse to take advantage of the private information she is offered, and seeks safer investment opportunities in the risk-free market. In a similar fashion, from Fig. 2(b), we can see that as the initial wealth level of the economic agent increases, the agent is expected to ignore the private information and turn her attention in the risk-free asset. This behaviour, which in the current framework is a byproduct of the exponential utility function (4.1), is in contrast with the behaviour of the investor in Liu et al. (2010); In their setting, under power-type preferences, the more wealthy the agent the more valuable is the information. On the other hand, in our setting, the more wealthy the manager is, the less is interested in the additional information and the risky asset, and enjoys the risk-free interest rate. Fig. 2. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths for various values of the parameter $$\gamma$$. In this case we consider $$\psi=0.5$$ and $$\lambda=0.2$$. (b) Average path of 6000 optimal investment strategy paths for various values of the initial wealth. In this case we consider $$\psi=0.5$$ and $$\lambda=0.2$$. Fig. 2. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths for various values of the parameter $$\gamma$$. In this case we consider $$\psi=0.5$$ and $$\lambda=0.2$$. (b) Average path of 6000 optimal investment strategy paths for various values of the initial wealth. In this case we consider $$\psi=0.5$$ and $$\lambda=0.2$$. Finally, from Fig. 3(a,b), we have the following interesting results: Fig. 3. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths for various values of the initial level of the Ornstein–Uhlenbeck stochastic factor. In this case we consider $$\psi=0$$ and $$\lambda=0.3$$. (b) Average path of 6000 optimal investment strategy paths for various values of parameter $$\theta$$ of the Ornstein–Uhlenbeck stochastic factor. In this case we consider $$\psi=0.3$$ and $$\lambda=0.2$$. Fig. 3. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths for various values of the initial level of the Ornstein–Uhlenbeck stochastic factor. In this case we consider $$\psi=0$$ and $$\lambda=0.3$$. (b) Average path of 6000 optimal investment strategy paths for various values of parameter $$\theta$$ of the Ornstein–Uhlenbeck stochastic factor. In this case we consider $$\psi=0.3$$ and $$\lambda=0.2$$. From Fig. 3(a), we can see that as the initial level of the stochastic factor process increases, the economic agent is expected to invest more in the risk-free asset. In fact, this is explained as follows. When the financial market evolves according to the stochastic differential equations (4.7) and (4.8), as the level of the stochastic factor increases, together with the market risk premium, the systematic risk associated with the risky asset (4.7) increases too, but the Sharpe ratio of the portfolio is always the same. In other words, the manager is not properly compensated for the additional risk she is willing to undertake. This makes the risky market (4.7) and (4.8) not a reasonable investment. Hence, investing in the risk-free asset seems the best alternative in this case. Additionally, we observe (by taking into account Fig. 1(a)) that even a value of $$\psi=0$$ leads to a more aggressive position in the risky market, compared to the symmetric information case. From Fig. 3(b), we can see that as the parameter $$\theta$$ increases, the economic agent turns her attention in the risk-free asset. In the current example, the stochastic factor evolves according to an Ornstein–Uhlenbeck process and $$\theta$$ is the long-term mean. A high value of the parameter $$\theta$$ means higher values for the stochastic factor, leading to more investment in the risk-free asset (according to the previous explanation). Remark 5.2 The effect of the volatility parameters $$\sigma$$ and $$\beta$$ is the same with the effect of $$\gamma$$, hence their numerical study is committed. 6. Conclusions In the present article, we studied an optimal control problem for a general stochastic factor model under the existence of some private information. To be more precise, we considered a portfolio manager, who has the possibility to invest part of her wealth in a financial market consisting of a risk-free asset and a risky one, whose coefficients depend on some exogenous correlated diffusion process, known as the stochastic factor. Additionally, the manager, from the very beginning of the trading interval, observes some information signal concerning the future. However, this signal is not precise in the sense it is subject to some observation noise. Within this very general framework, by resorting to the mixture of dynamic programming and initial enlargement of filtrations techniques, we effectively attacked an expected utility maximization problem by taking into account the enlarged information set of the manager. In this vein, we provided a general result characterizing the value function of the problem at hand and the optimal control law, by solving the associating Hamilton–Jacobi–Bellman equation. Moreover, in the special case, when: (i) the manager operates under exponential preferences, (ii) the stochastic factor process evolves according to an Ornstein–Uhlenbeck process and (iii) the information signal concerns a future noisy value of the Brownian motion implying uncertainty in the stock price process, we provided closed form solutions for the optimal investment decision and the optimal value function. In addition, by employing an Euler–Maruyama scheme followed by a Monte-Carlo approach, we numerically studied this special example, capturing in this way the qualitative and quantitative features of the private information on the optimal investment strategy. The model is extended in Supplementary Material in two major ways: (i) we solve the proposed model for other utility function, namely, in the case of the logarithmic and the power utilities and (ii) we present the explication of our framework in the multi-dimensional case, that is, when the market is consisted of multiple risky assets and multiple stochastic factors. Supplementary Material Supplementary Material is available at http://www.imaman.oxfordjournals.org/. Acknowledgements The authors would like to thank the editor and the three anonymous referees for their constructive comments that led to an improvement of this work. Part of this research was conducted while the first author was a Postdoctoral Research Fellow at the Department of Statistics of the Athens University of Economics and Business is gratefully acknowledged. Funding I.B. was supported by the research funding program: “Research Funding at Athens University of Economics and Business for Excellence and Extroversion: Action 2”: EP-2448-01/01-01, EP-2448-01/01-02, duration: 18/02/2016-31/01/2017. Footnotes 1Fei & Wu (2003) study an optimal investment problem under the existence of some private information in the case of a large investor whose actions affect the market prices. 2Which requires the conditional distribution of $$\psi$$ given $$\mathscr{F}_{t}$$ to be absolutely continuous to the law of $$\psi$$. 3For example, one could consider the specific case $$\psi = \lambda W(T_{0})+(1-\lambda)\epsilon$$, with $$T_{0}>T$$, where $$\epsilon$$ is a standard normal random variable which is independent of $$W$$. 4For a further investigation of the relationship between the additional information available and the information drift in a general setting, we refer the interested reader to the article of Amendinger & Imkeller (2007). 5These arguments require the function $$\hat{u}$$ to satisfy a quadratic growth condition. 6Under the additional assumption that the coefficients of the market are bounded by some large constant and by taking into account equation (2.2). 7A similar result can also be found in Zariphopoulou (2009), who studied an optimal investment problem within a stochastic factor model under symmetric information. 8This can be seen from the fact that $$-\frac{u^{''}(x)}{u^{'}(x)}=\gamma$$. 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( 2014 ) Optimal proportional reinsurance and investment problem with jump-diffusion risk process under effect of inside information, Front. Math. China , 9 , 965 – 982 . Google Scholar CrossRef Search ADS Zariphopoulou T. ( 2001 ) A solution approach to valuation with unhedgeable risks, Finance Stoch. , 5 , 61 – 82 . Google Scholar CrossRef Search ADS Zariphopoulou T. ( 2009 ) Optimal asset allocation in a stochastic factor model–-an overview and open problems, Advanced Financial Modelling, Radon Series Comp. Appl. Math. , 8 , 427 – 453 . © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Management Mathematics Oxford University Press

Portfolio management in a stochastic factor model under the existence of private information

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1471-678X
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Abstract

Abstract In the present article, we study an optimal control problem for a general stochastic factor model under the existence of private information. More precisely, we consider a portfolio manager who has the possibility to invest part of her wealth in a financial market consisting of a risk-free asset and a risky one, whose coefficients depend on some external stochastic factor. Moreover, we assume that the manager, from the beginning of the trading interval, observes an information signal associated with the future evolution of the risky asset. This information is not clear but is subject to some observation noise. Within a very general framework, by resorting to a mixture of dynamic programming and initial enlargement of filtrations techniques, we characterize the optimal value function and the feedback control law, by solving an expected utility maximization problem under the enlarged information set of the economic agent. In the case of the exponential utility function and considering a specific form for the market parameters and the information signal, we provide closed form solutions for the optimal investment decision and the optimal value function. Additionally, by employing an Euler–Maruyama scheme followed by a Monte-Carlo approach, we numerically study the impact of the private information on the optimal investment strategy for this concrete example. The article has Supplementary Material, which provides the extension of our model to other classes of utility functions (logarithmic and power) and also presents the general case of multiple assets and factors. 1. Introduction Stochastic optimal control theory has found itself to establish a ubiquitous presence in many fields such as mathematical economics and finance. It is the backbone of most of the modern theory of portfolio management, policy design in macroeconomics, international banking or environmental economics. While heavily used in applications, it is also a theory very rich from the mathematical point of view, which still presents very important open problems. To the best of our knowledge, the implementation of stochastic optimal control techniques on financial related problems was originated in the pioneering works of Merton (1969, 1971), who studied a continuous time investment and consumption problem in a financial market consisting of two assets: a risk-free asset and a risky one whose dynamics are described by the geometric Brownian motion model. For the special case where the economic agent operates under constant relative risk aversion preferences, Merton (for more information, see Merton, 1969) was able to provide closed form solutions for the value function, the optimal investment strategy and the optimal consumption rules. Since then, many papers and books that bridge stochastic optimal control techniques and mathematical finance have been written (see, among others, Browne, 1995; Duffie et al., 1997; Zariphopoulou, 2001, 2009; Pham, 2009 and references therein). One of the most salient features of the mentioned theory, is the assumption of a common public information flow, according to which all the economic agents that operate in a financial market make their portfolio (investment and/or consumption) decisions. This information is expressed in terms of the natural filtration which is generated by the stochastic process implying uncertainty in the market, for example a Brownian motion, and is available at everyone’s disposal from the very beginning of the trading interval. This is achieved by postulating that all portfolio strategies are non-anticipating, that is, are progressively measurable with respect to this filtration. However, research and empirical results have provided evidence that this assumption is not fully valid. Niederhoffer & Osborne (1966) pointed out that there exist specialists operating on major security exchanges that have access to some private information concerning the unexecuted limit orders and this information can be used for certain profit purposes. In this vein, Fama (1970) empirically proved that financial markets are not strongly efficient in the sense that private information is not fully reflected in security prices. Ipso facto, the existence of asymmetrically informed economic agents in financial markets is broadly accepted and modelling their behaviour has been one of the most challenging concepts in mathematical finance. A very important and more general class of models of asymmetric information is the inside information models, according to which in a financial market operate two type of agents. The honest economic agents (or regulars) whose portfolio decisions are based on the public information flow which is induced by the market noises and the informed economic agents (or insiders) who exclusively possess some future information concerning, for example, the evolution of the market price movements in addition to the public information flow. Depending when the additional private information is available at the insider’s disposal, there are two sophisticated techniques to effectively treat inside information models, namely, initial (when the additional information is available from the very beginning of the trading interval) and progressive (when the private information is progressively known, that is, as time passes) enlargement of filtrations. By enlargement of a filtration we mean the expansion of a given filtration $$\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$$ to a new filtration $$\mathbb{G}=(\mathcal{G}_{t})_{t\geq 0}$$, defined so as to satisfy the usual hypotheses of right continuity and completeness and $$\mathcal{F}_{t}\subset\mathcal{G}_{t}$$ for each $$t\geq 0$$. According to initial enlargement of filtrations theory, the additional information is stored in a random variable and is available at the insider’s disposal from the beginning of the trading interval. The natural filtration of the market is then enlarged with the sigma algebra which is generated by this random variable and results to the filtration of the insider. This technique has proven itself to be a very efficient tool for the study of the problem of inside information in a variety of cases (see for example, Pikovsky & Karatzas, 1996; Amendinger et al., 1998; Grorud & Pontier, 1998; Imkeller et al., 2001; Fei & Wu, 2003; Amendinger & Imkeller, 2007; Danilova et al., 2010; Kohatsu-Higa & Ortiz-Latorre, 2010; Liu et al., 2010; Hansen, 2013), for certain problems arising in mathematical finance and also (Baltas et al., 2012; Xiong et al., 2014) for an inside information model in the insurance/reinsurance market. In mathematical finance, a sound mathematical model which has been heavily applied to describe the evolution of a stock price process, is the geometric Brownian motion model. This model has its origins in the inspiring work of Samuelson (1965) and acted as point of reference for the celebrated, Nobel prize awarded, Black–Scholes model. Since then, this model has risen to fame and has been extensively studied in many works (see e.g. Shreve & Soner, 1994; Browne, 1995; Duffie et al., 1997 and references therein). However, it fails to capture the effect of many important exogenous market parameters. To the best of our knowledge, the first attempt in extending this model, is the article of Karatzas et al. (1991) who assume the coefficients to be deterministic functions of time. A perhaps more realistic and much general framework is provided by the so called stochastic factor models. Substantially, the main characteristic of such a model is that the prices of the underlying risky asset are allowed to be functions of an another stochastic process, called the stochastic factor, which evolves according to a correlated diffusion process. This general setting covers, among others, the interesting case of stochastic volatility and stochastic interest rate models. Within this framework, there exist many papers and books of both mathematical and practical interest (see, among others, Kim & Omberg, 1996; Fouque et al., 2000; Bielecki & Pliska, 2001; Zariphopoulou, 2001, 2009; Liu, 2007; Delong & Klüppelberg, 2008 and references therein). This work aims to contribute to the existing theory of portfolio management, in complex environments, under the existence of private information. The novelty of our work lies in the fact that, for the very first time in the relative literature, we expand the setting of general stochastic factor models to include the effect of private information in the study of a related optimal investment problem. To be more precise, we consider a portfolio manager who is endowed with some initial wealth and is allowed to invest in a financial market consisting of a risk-free asset (bond or bank account) and a risky one, whose evolution depends on some exogenous economic factor. This factor, evolves stochastically according to a correlated diffusion process with general coefficients. Additionally, the manager possesses, from the very beginning of the trading interval, some additional information concerning the future, which stems from the observation of a private signal, thus introducing in this way inside information aspects to our model. However, this information is not precise, in the sense that is subject to some observation noise. The main focus of interest is to study a general expected utility maximization problem by resorting to a rigorously defined mixture of dynamic programming and initial enlargement of filtrations technique so as to reduce the calculation of the optimal portfolio decision to the solution of a relevant Hamilton–Jacobi–Bellman equation valued for a wide class of utility functions. In this vein, we provide a general result characterizing the value function of the problem at hand and the optimal control law. In the special case when the economic agent operates under exponential preferences, we provide closed form solutions for the value function and the optimal investment decision. In addition, we present a fully worked example by considering a very specific form for the additional information and the market parameters, namely, when the stochastic factor evolves according to an Ornstein–Uhlenbeck process and the information signal concerns a future value of the underlying Brownian motion, driving uncertainty in the risky asset. By employing a mixture of an Euler–Maruyama scheme and a Monte-Carlo approach, we are able to effectively study numerically, in full detail and for the whole trading interval (this is, to the best of our knowledge, among the first papers to do so), the optimal investment strategy of the insider capturing in this way the qualitative and quantitative nature of her behaviour. Additionally, in the associated Supplementary Material, we extend our framework in two major ways: (i) we solve the proposed model in the case of the logarithmic and the power utility functions and (ii) we present the multi-dimensional case with multiple risky assets and stochastic factors. The present article is organized as follows. In Section 2 we describe our model, and in Section 3 we formulate the problem of maximizing the insider’s expected utility from terminal wealth and give a general result characterizing the value function and the optimal investment strategy, by solving the corresponding Hamilton–Jacobi–Bellman equation. Section 4 provides closed form solutions for the value function and the optimal investment strategy, when the economic agent operates under exponential preferences and additionally we provide a fully worked example by considering a specific form for the private information and the market parameters. Finally, in Section 5, we numerically study the results of the specific example of Section 4. The article has Supplementary Material that extends our results by considering (i) other utility functions (logarithmic and power type preferences) and (ii) the case of multiple assets and factors. 2. The model Let us consider the filtered probability space $$(\Omega, \mathscr{F}_{t}, \mathbb{P})$$ that satisfies the usual hypotheses of right continuity and completeness, where $$\mathscr{F}_{t}=\sigma(W(s), B(s)\,, s\leq t)$$ is the natural filtration induced by the standard independent Brownian motions $$\left\{W(t), \, t\geq 0\right\}$$ and $$\left\{B(t), \, t\geq 0\right\}$$. 2.1. The financial market We adopt a model for the financial market on the fixed finite time horizon $$[0,T]$$, with $$T\in(0,\infty)$$, consisting of a risk-free asset (bond or bank account) with dynamics described by \begin{equation} \begin{aligned} dP(t) &= rP(t)dt \\ P(0) &=1, \end{aligned} \end{equation} (2.1) where $$r>0$$ denotes the interest rate and a risky one (stock or index) with dynamics given by \begin{equation} \begin{aligned} \frac{dS(t)}{S(t)}&= \Big[r +\mu(Y(t))\Big]dt + \sigma(Y(t))dW(t)\\ S(0)&=S_{0}>0, \end{aligned} \end{equation} (2.2) where $$P(t)$$ denotes the price of the bond and $$S(t)$$ the price of the stock at time $$t\in[0,T]$$. The stochastic process $$\left\{Y(t),\, t\geq 0\right\}$$ may be interpreted as an external economic factor that has an impact on the prices of the risky asset and satisfies the following stochastic differential equation \begin{equation} \begin{aligned} dY(t)&=\alpha(Y(t))dt + \beta(Y(t))\Big[\rho dW(t) + \sqrt{1-\rho^{2}}dB(t)\Big]\\ Y(0)&=y \in\mathbb{R}, \end{aligned} \end{equation} (2.3) where $$\rho\in[-1,1]$$ stands for the correlation coefficient. We assume that the stochastic factor cannot be directly traded leading to the incompleteness of the market, (as long as $$\rho^{2}\neq 1$$), since the economic agent faces more sources of uncertainty than traded assets. This very general framework has been heavily studied in the relative literature (see e.g. Zariphopoulou, 2009) and allows for the effective modelling of the influence that many important exogenous factors, of either macroeconomic or microeconomics nature, have, on the evolution of the stock prices. Assumption 2.1 The market coefficients $$\ell=(\mu,\sigma,\alpha,\beta)$$ in equations (2.2) and (2.3), satisfy the following conditions (i) The function $$\ell$$ is sublinear and globally Lipschitz, that is, for every $$y,\bar{y}\in\mathbb{R}$$ there exists some constant $$C>0$$, such that \begin{equation*} |\ell(y)-\ell(\bar{y})|\leq C|y-\bar{y}|, \end{equation*} and \begin{equation*} \ell^{2}(y)\leq C(1+y^{2}). \end{equation*} (ii) For every $$y\in\mathbb{R}$$ there exist constants $$c_{1}$$ and $$c_{2}$$, such that \begin{align*} \sigma(y)\geq c_{1}>0,\quad {\rm and}\quad \beta(y)\geq c_{2}>0. \end{align*} (iii) The solution of the stochastic differential equation (2.2) satisfies $$S(t)\geq 0$$ for every $$t\in[0,T]$$ and initial condition $$S(0)>0$$. Remark 2.1 The conditions of Assumption 2.1 are somewhat standard in the relative literature (see e.g. Zariphopoulou, 2001, 2009) and can be relaxed in a number of ways depending on the underlying framework. In our context, it is straightforward to verify that under Assumption 2.1(i) the system of stochastic differential equations (2.2) and (2.3) admits a unique strong solution. For more information on this subject we refer the interested reader to Chapter VI of Arnold (1974). The non-degeneracy Assumption 2.1(ii) guarantees the well-posedness of the associating value function (cf. equation (3.1)) and of the resulting partial differential equation (cf. equation (3.3)). 2.2. Stochastic differential equation for the manager’s wealth We envision a portfolio manager, who, at time $$t=0$$, is endowed with some initial wealth $$x>0$$ and whose actions cannot affect the market prices.1 The portfolio process $$\pi(t)=\pi(t,\omega):[0,T]\times\Omega\rightarrow \Pi\subset\mathbb{R}$$ denotes the proportion of her wealth $$X^{\pi}(t)$$ invested in the risky asset. The remaining proportion $$(1-\pi(t))X^{\pi}(t)$$ is invested in the risk-free asset. Here, $$\Pi$$ is a fixed closed and convex subset of $$\mathbb{R}$$; typically compact. As a result, the wealth process corresponding to the strategy $$\pi(t)$$, is defined as the solution of the following stochastic differential equation \begin{equation*} dX^{\pi}(t) = \pi(t)X^{\pi}(t)\frac{dS(t)}{S(t)} + \left(1-\pi(t)\right)X^{\pi}(t)\frac{dP(t)}{P(t)}. \end{equation*} Therefore, \begin{equation} \begin{aligned} \frac{dX^{\pi}(t)}{X^{\pi}(t)} &= \left[r + \pi(t)\mu(Y(t))\right]dt + \pi(t)\sigma(Y(t))dW(t) \\ X^{\pi}(0)&=x>0. \end{aligned} \end{equation} (2.4) Definition 2.1 Let $$\mathbb{F}$$ be a general filtration. We denote by $$\mathcal{A}({\mathbb{F}};T)$$ the class of admissible strategies $$\pi(t)$$ that satisfy the following conditions: (i) $$\pi(t):[0,T]\times\Omega\rightarrow \Pi\subset\mathbb{R}$$ is a progressively measurable mapping with respect to the filtration $$\mathbb{F}$$; (ii) $$\mathbb{E}\left[\displaystyle\int_{0}^{T} \left(\sigma(Y(t))\pi(t)\right)^{2}dt \right]< \infty,\,\, \mathbb{P}$$-a.s.; (iii) The SDE (2.4) admits a unique strong solution, denoted by $$X^{\pi}(t)$$. 2.3. Inside information Suppose now that the portfolio manager, in addition to the publicly available information flow $$\mathbb{F}=(\mathscr{F}_{t})_{t\in[0,T]}$$, possesses, from the very beginning of the trading interval $$[0,T]$$, some private information through the observation of a Brownian signal, concerning the stochastic process $$\left\{W(t),\, t\geq 0\right\}$$ introducing uncertainty in the stock price process (2.2), at the future time $$T_{0}\geq T$$. This additional information is stored in a certain $$\mathscr{F}_{T_{0}}$$-measurable random variable $$\psi$$, which is available only at her disposal and hidden from any other economic agents that presumably operate on the same financial market. Consequently, the informed portfolio manager (also called insider) can choose her strategy from the class of $$\mathbb{G}$$ adapted processes, where the filtration $$\mathbb{G}=(\mathscr{G}_{t})_{t\in[0,T]}$$ is given by the augmentation of the natural filtration (which is available at everyone’s disposal) with the inside information, represented by the $$\sigma$$-algebra generated by $$\psi$$, that is, \begin{equation*} \begin{aligned} \mathscr{G}_{t} &= \mathscr{F}_{t} \vee \sigma(\psi) \\ &= \sigma(W(s), B(s)\,, s\leq t) \vee \sigma(\psi),\,\, t\in[0,T], \end{aligned} \end{equation*} and is defined so as to satisfy the usual hypotheses of right continuity and completeness. Obviously, $$\mathbb{F} \subset \mathbb{G}$$, hence the insider can choose her portfolio from a larger class of admissible strategies than any other economic agent. Typically, the core of initial enlargement of filtrations theory, dictates, that if the random variable $$\psi$$ satisfies the famous Jacod’s Hypothesis2 (see e.g. Jacod, 1985, Chapter V of Jeanblanc et al., 2009 and Kohatsu-Higa & Ortiz-Latorre, 2010 or Chapter I of Mansuy & Yor, 2006), then, $$W$$ becomes a $$(\mathbb{G}, \mathbb{P})$$-semimartingale with respect to the enlarged filtration. In fact, we have the following well-known result, a version of which can be found in Danilova et al. (2010). Lemma 2.1 Suppose $$\psi$$ is an $$\mathscr{F}$$-measurable random variable, such that it admits a conditional density $$g:[0,T]\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{+}$$ with respect to $$\mathbb{F}$$, so that for any test function $$h:\mathbb{R}\rightarrow\mathbb{R}$$ \begin{equation} \mathbb{E}\Big[h(\psi)|\mathscr{F}_{t}\Big] = \int_{\mathbb{R}}h(r)g(t,r,W(t))dr,\,\, t\in[0,T]. \end{equation} (2.5) Suppose further that $$g(t,r,z)$$ is smooth enough in $$z$$ and it satisfies \begin{equation} \int_{0}^{t}\Big|\frac{g_{z}(s,r,z)}{g(s,r,z)}\Big|ds<\infty\,\,\,\,\text{and}\,\,\,\,\int_{\mathbb{R}}\big|g_{z}(t,r,z)\big|dr<\infty, \end{equation} (2.6) for a.e. $$z\in\mathbb{R}$$ and $$t\in[0,T]$$. Then, the semimartingale decomposition of $$W$$ with respect to $$\mathbb{G}$$ is given by \begin{equation} W(t) = \widetilde{W}(t) + \int_{0}^{t}\frac{g_{z}(s,\psi,W(s))}{g(s,\psi,W(s))}ds,\,\,\,t\in[0,T], \end{equation} (2.7) where $$\widetilde{W}$$ is a $$(\mathbb{G}, \mathbb{P})$$-Brownian motion. The stochastic process \begin{equation} \frac{g_{z}(t,\psi,W(t))}{g(t,\psi,W(t))},\,\,\,t\in[0,T], \end{equation} (2.8) is referred to as the information drift. Proof. For a detailed proof, see e.g. Danilova et al. (2010) and Chapter I of Mansuy & Yor (2006). □ In the present article, we assume that knowledge of the value of $$\psi$$ allows the manager to infer information concerning $$W(T_{0})$$, which however is not explicitly specified at this point. This additional information will serve as a means to predict the movement of stock prices and will possibly help her to obtain a better position in the market described by equations (2.1–2.3). A measure of the efficiency of $$\psi$$ will be denoted by a scalar parameter $$\lambda\in[0,1]$$. To wit, a value of $$\lambda=1$$ implies that the information signal is precise whereas a value of $$\lambda=0$$ implies a completely imprecise information signal.3 For various forms of the inside information, Lemma 2.1 gives the semimartingale decomposition of the stochastic process $$W$$ under the enlarged filtration $$\mathbb{G}$$. As in general the conditional density $$g$$ corresponding to $$\psi$$ is not known, motivated by Lemma 2.1, we adopt the following semimartingale decomposition for $$W$$ under the enlarged filtration $$\mathbb{G}$$, \begin{equation} W(t) = \widetilde{W}(t) + \int_{0}^{t}m(s)ds,\,\,\,t\in[0,T], \end{equation} (2.9) where the stochastic process $$m(t):=m(t,W(t);T_{0},\psi,\lambda)$$ stands for the information drift. Such a choice is not irrational, as equation (2.8) already dictates that the information drift is a stochastic process that depends on the information signal $$\psi$$ and the stochastic process $$W(t)$$. Additionally, due to the assumption on the observed signal, we have to take into account two more parameters associated with the signal, namely $$\lambda$$ and $$T_{0}$$. At this point, and before proceeding any further in our analysis, it is crucial to make the following assumption, as it partially guarantees the applicability of the classical theory (see e.g. Karatzas, 1989; Karatzas & Shreve, 1998). Assumption 2.2 The information drift satisfies the integrability condition \begin{equation*} \mathbb{E}\left[\int_{0}^{T}m^{2}(s)ds\right]<\infty. \end{equation*} Clearly, from equation (2.9) it follows that the stochastic process $$m(t)$$ must be $$\mathscr{G}_{t}$$-measurable. However, in order to make further progress to the stochastic optimal control problem we have in mind, we have to restrict to more specific forms for the information drift. For particular classes of initial enlargements related to Brownian signals, it can be shown (cf. Proposition 4.1 and especially equation (4.11) in view of equation (4.10) and equation (4.24) in view of equation (4.23), see also Danilova et al., 2010; Hansen, 2013) that a plausible assumption for the dynamics of $$m(t)$$ is the following. Assumption 2.3 The information drift evolves according to the following stochastic differential equation \begin{equation} \begin{aligned} dm(t) &= f(t)d\widetilde{W}(t) \\ m(0) &=m_{0}(T_{0},\psi,\lambda)\in\mathbb{R}, \end{aligned} \end{equation} (2.10) where $$f(t):=f(t;T_{0},\lambda)$$ is a square integrable deterministic function, for every $$t\in[0,T]$$. Remark 2.2 At this point, let us comment a little bit the form of the functions appearing in equations (2.9) and (2.10): (i) $$m(t)$$ is the information drift which is generated by the information signal $$\psi$$. This information concerns the $$\left(\mathbb{F}, \mathbb{P}\right)$$-Brownian motion $$W$$, at the future time $$T_{0}$$. Furthermore, it is linked with a scalar parameter $$\lambda$$, denoting its precision level. Hence, it seems natural to let the information drift be a function of $$W$$ and depend on the parameters $$T_{0}, \lambda$$ and $$\psi$$4. (ii) $$f(t)$$ aims to capture the dynamics of the information drift in the enlarged filtration. For the special cases we have in mind, it is obtained by an application of Itô’s lemma to the information drift (assuming, of course, enough smoothness). This approach suggests that $$f(t)$$ takes into account the information horizon $$T_{0}$$ and the parameter $$\lambda$$, and not $$W$$ or $$\psi$$ (iii) $$m(0)$$ is the value of the information drift at time $$t=0$$. Assuming that $$W$$ is a standard Brownian motion (i.e. $$W(0)=0$$), it is natural to expect $$m(0)$$ to be a function of $$T_{0}, \lambda$$ and $$\psi$$. Our intention from now on is to plug the semimartingale decomposition (2.9), by taking into account equation (2.10), in the stochastic differential equation (2.4) that describes the manager’s wealth at time $$t$$, so as to reflect the additional information she possesses. Proposition 2.1 The wealth of the informed economic agent at time $$t\in[0,T]$$, taking explicitly into account the information signal $$\psi$$ she observes at time $$t=0$$, is given by the system of stochastic differential equations \begin{align} \begin{split} \frac{dX^{\pi}(t)}{X^{\pi}(t)}&=\Big[r + \pi(t)\mu(Y(t)) + \pi(t)\sigma(Y(t))m(t)\Big]dt + \pi(t)\sigma(Y(t))d\widetilde{W}(t)\\ dY(t)&= \Big[\alpha(Y(t)) + \rho\beta(Y(t))m(t)\Big]dt + \beta(Y(t))\Big[\rho d\widetilde{W}(t) + \sqrt{1-\rho^{2}}dB(t)\Big]\\ dm(t)&= f(t) d\widetilde{W}(t), \end{split} \end{align} (2.11) with initial condition $$(X^{\pi}(0), Y(0), m(0))=(x,y,m)\in\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}.$$ Proof. The proof follows by substituting equation (2.9) in the stochastic differential equation (2.4) that describes the wealth of the economic agent under symmetric information and in the stochastic differential equation (2.3) that describes the evolution of the external factor process. □ 3. The optimization problem In this section, we formally state the problem of maximizing the expected utility from terminal wealth for the informed economic agent by taking explicitly into account the additional information she possesses. 3.1. The problem We are now in a position to rigorously state the problem. Let $$\mathcal{A}({\mathbb{G}};T)$$ denote the admissible class of the insider taking into account her additional information, as it follows in terms of Definition 2.1. Assumption 3.1 The utility function $$U:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$$ is a function that is strictly increasing, strictly concave, $$C^{1}$$, and satisfies the Inada conditions \begin{align} \lim_{x\rightarrow +\infty}U^{'}(x) &= 0 \nonumber\\ \lim_{x\rightarrow 0}U^{'}(x) &= +\infty \, \Big(\text{or} \lim_{x\rightarrow -\infty}U^{'}(x) = +\infty\Big).\nonumber \end{align} Remark 3.1 Our aim is to enlarge the state space with two additional variables. The variable $$y$$ which corresponds to the stochastic factor process, and the variable $$m$$ which corresponds to the private information. However, one should emphasize that: (i) $$m(t)$$ is adapted to the observed (enlarged) filtration $$\mathbb{G}$$, hence $$m(t)$$ is known to the informed agent at any time $$t\in[0,T]$$, once the value of the information signal $$\psi$$ is available (ii) the stochastic factor cannot be directly traded, but it is assumed to be also observed at any time $$t\in[0,T]$$. To state differently, the economic agent knows the state of $$(y,m)\in\mathbb{R}\times\mathbb{R}$$ when choosing the optimal control. As a result, we provide the necessary Markovian structure and we are able to apply dynamic programming techniques. Our aim is to maximize the expected utility of the terminal wealth of the informed economic agent, by taking into account the additional information she possesses, that is, we face the problem \begin{equation} u(t,x,y,m) = \sup_{\pi\in\mathcal{A}({\mathbb{G}};T)} \mathbb{E}\Big[U\left(X^{\pi}(T)\right) \big| \psi \Big], \end{equation} (3.1) subject to the dynamic constraint (2.11) and with initial condition $$\left(X^{\pi}(t), Y(t), m(t)\right) = (x,y,m)\in\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}$$. 3.2. Hamilton–Jacobi–Bellman equation of insider’s wealth In order to solve the problem (3.1) and explore the Markovian structure of our model in the enlarged state space representation, given by equation (2.11), we use dynamic programming techniques to derive the associated Hamilton–Jacobi–Bellman equation for the value function. We will employ the following notation. Let $$z=(x,y,m)\in\mathbb{R}_{+}\times\mathbb{R}^{2}$$ and $$t\in[0,T]$$ and consider functions $$u:\mathbb{S}\equiv[0,T]\times\mathbb{R}_{+}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$$ with values $$u(t,z) = u(t,x,y,m)$$. As usual, subscripts with respect to a variable denote partial differentiation of the function with respect to this variable. We will also denote by $$\mathcal{C}^{1,2}$$ the space of functions $$u$$, as above, which are once continuously differentiable with respect to the time variable and twice continuously differentiable with respect to the space variable. The generator for the state process (2.11) subject to the choice $$\pi(t)$$ admits the form \begin{align*} \mathcal{L}^{\pi}h(t,x,y,m) &= \Big[r+ \pi\mu(y) + \pi\sigma(y)m\Big]xh_{x} + \Big[\alpha(y) + \rho\beta(y)m\Big]h_{y}\nonumber\\ &\quad + \frac{1}{2}\Big[ \pi^{2}x^{2}\sigma^{2}(y)h_{xx}+\beta^{2}(y)h_{yy}+f^{2}(t)h_{mm}\Big]\\ &\quad +\pi x \sigma(y)f(t)h_{xm} + \rho\beta(y)f(t)h_{ym} + \pi x \sigma(y)\rho\beta(y)h_{xy}, \nonumber \end{align*} where $$h\in\mathcal{C}^{1,2}$$. Thus, if $$u\in \mathcal{C}^{1,2}$$ one can show that the value function $$u$$ satisfies the following Hamilton–Jacobi–Bellman equation \begin{equation} \begin{aligned} u_{t} + \sup_{\pi\in\mathcal{A}({\mathbb{G}};T)}\mathcal{L}^{\pi}u(t,x,y,m) &= 0\\ u(T,x,y,m) &=U(x). \end{aligned} \end{equation} (3.2) As a result of the state space enlargement we have introduced, we are able to characterize the value function and the optimal investment strategy in terms of the solution of an appropriate Hamilton–Jacobi–Bellman equation. Theorem 3.1 Consider the fully non-linear partial differential equation \begin{equation} \begin{aligned} u_{t}&+rxu_{x}+\alpha(y)u_{y}-\frac{1}{2}\left[\frac{\mu(y)}{\sigma(y)}+m\right]^{2}\frac{u_{x}^{2}}{u_{xx}} +\frac{1}{2}\beta^{2}(y)\left[u_{yy}-\rho^{2}\frac{u_{xy}^{2}}{u_{xx}}\right] +\frac{1}{2}f^{2}(t)\left[u_{mm}-\frac{u_{xm}^{2}}{u_{xx}}\right]\\ &+ \rho\beta(y)m\left[u_{y}-\frac{u_{x}u_{x y}}{u_{xx}}\right]+\rho\beta(y)f(t)\left[u_{ym}-\frac{u_{xm}u_{x y}}{u_{xx}}\right]-\rho\beta(y)\frac{\mu(y)}{\sigma(y)}\frac{u_{x}u_{xy}}{u_{xx}}\\ &-f(t)\left[\frac{\mu(y)}{\sigma(y)}+m\right]\frac{u_{x}u_{xm}}{u_{xx}}=0. \end{aligned} \end{equation} (3.3) If (3.3) admits a solution $$\hat{u}\in\mathcal{C}^{1,2,2,2}(\mathbb{S})$$ and furthermore the function \begin{equation} \pi^{*}(t,x,y,m)=\left\{-\left[\frac{\mu(y)}{\sigma(y)} +m\right]\frac{\hat{u}_{x}}{\hat{u}_{xx}}-\rho\beta(y)\frac{\hat{u}_{xy}}{\hat{u}_{xx}} - f(t)\frac{\hat{u}_{xm}}{\hat{u}_{xx}}\right\}\frac{1}{x\sigma(y)}, \end{equation} (3.4) provides an admissible control law, then $$\hat{u}$$ is the value function for the control problem and $$\pi^{*}$$ is the optimal feedback control law. Proof. For each quadruple $$(t,x,y,m)\in\mathbb{S}$$ we must solve the following equation \begin{eqnarray} && u_{t}+ rxu_{x} + \Big[\alpha(y) + \rho\beta(y)m\Big]u_{y} + \frac{1}{2}\Big[\beta^{2}(y)u_{yy} + f^{2}(t)u_{mm}\Big]+ \rho\beta(y)f(t)u_{ym} \notag\\[-9pt]\\[-3pt] &&\quad+\max_{\pi} \left\{ \Big[\pi\mu(y) + \pi\sigma(y)m\Big]xu_{x}+\frac{1}{2}\pi^{2} x^{2} \sigma^{2}(y)u_{xx} +\pi x\sigma(y)f(t)u_{xm} + \pi x \sigma(y)\rho\beta(y) u_{xy} \right\}=0,\notag \end{eqnarray} (3.5) with boundary condition $$u(T,x,y,m)=U(x)$$. Assume that the maximum in equation (3.5) is attained in the interior of the control region (assuming of course that the control set $$\Pi$$ has a non-empty interior $$\Pi^{0}$$), that is, $$\pi^{*}(t,x,y,m)\in \Pi^{0}$$ for all $$(t,x,y,m)\in\mathbb{S}$$. Differentiating the above expression with respect to $$\pi$$ and setting the derivative equal to zero, gives the candidate optimal control \begin{equation} \hat{\pi}(t,x,y,m)=\left\{-\left[\frac{\mu(y)}{\sigma(y)} +m\right]\frac{u_{x}}{u_{xx}}-\rho\beta(y)\frac{u_{xy}}{u_{xx}} - f(t)\frac{u_{xm}}{u_{xx}}\right\}\frac{1}{x\sigma(y)}. \end{equation} (3.6) In this case, if we place this expression back in equation (3.5), we arrive to the non-linear partial differential equation (3.3). Assume now that the partial differential equation (3.3) admits a classical solution $$\hat{u}\in\mathcal{C}^{1,2,2,2}(\mathbb{S})$$ that satisfies the differentiability conditions $$\hat{u}_{xx},\hat{u}_{yy},\hat{u}_{mm}<0$$ and $$\hat{u}_{x},\hat{u}_{y},\hat{u}_{m}>0$$. By substituting this solution back in equation (3.6) leads to equation (3.4). If moreover $$\hat{\pi}(t,x,y,m)\in \Pi^{0}$$, then $$\pi^{*}(t,x,y,m)$$ coincides with $$\hat{\pi}(t,x,y,m)$$. The rest is a straightforward application of the verification theorem and heavily relies on Itô’s lemma and standard arguments5. For a complete proof, we refer the interested reader to Chapter 3 of Pham (2009) □ Remark 3.2 Equation (3.3) may not in general admit smooth solutions, in which case it can be shown that the value function of the optimal control problem corresponds to a weaker notion of solution (viscosity solution) for which the verification theorem becomes more subtle (see e.g. Crandall et al., 1992 or Fleming & Soner, 2006). However, for a large class of utility functions of interest in applications, smooth solutions of (3.3) can be explicitly constructed (cf. Section 4 as well as the Supplementary Material), thus rendering Theorem 3.1 a useful tool in the solution of the corresponding optimal control problem. Remark 3.3 Consider the maximization problem $$\sup_{\pi\in\Pi}M(\pi)$$, where the function $$\pi\mapsto M(\pi)$$ is defined as \begin{equation} \begin{aligned} M(\pi):=\Big[\pi\mu(y) + \pi\sigma(y)m\Big]x\hat{u}_{x}+\frac{1}{2}\pi^{2} x^{2} \sigma^{2}(y)\hat{u}_{xx} +\pi x\sigma(y)f(t)\hat{u}_{xm} + \pi x \sigma(y)\rho\beta(y) \hat{u}_{xy}. \end{aligned} \end{equation} (3.7) If $$\hat{\pi}(t,x,y,m)\notin \Pi^{0}$$, then $$\pi^{*}(t,x,y,m)$$ is attained at some point at $$\partial \Pi$$, since, under the basis of our assumptions ($$\hat{u}_{xx}<0$$), the function $$M(\pi)$$ is strictly concave on the closed convex set $$\Pi$$. Let us for example consider the simple case $$\Pi=[0,1]$$. This case corresponds to no short-selling the risky asset or borrowing. Additionally, we define the sets \begin{equation} \begin{aligned} \mathfrak{U}_{1}&\equiv \left\{(t,x,y,m)\in\mathbb{S}: \hat{\pi}(t,x,y,m)\geq 1\right\}\\ \mathfrak{U}_{2}&\equiv \left\{(t,x,y,m)\in\mathbb{S}: \hat{\pi}(t,x,y,m)\leq 0\right\}\\ \mathfrak{U}_{3}&\equiv \left\{(t,x,y,m)\in\mathbb{S}: 0<\hat{\pi}(t,x,y,m)<1\right\}\!. \end{aligned} \end{equation} (3.8) Then, the optimal investment strategy is given by \begin{equation} \pi^{*}(t,x,y,m)=\begin{cases}1 &\,\, ;(t,x,y,m)\in\mathfrak{U}_{1}\\ \hat{\pi}(t,x,y,m) &\,\, ;(t,x,y,m)\in\mathfrak{U}_{3}\\ 0 &\,\, ;(t,x,y,m)\in\mathfrak{U}_{2},\end{cases} \end{equation} (3.9) and one should be able to check6 that $$\pi^{*}(t,x,y,m)\in\mathcal{A}({\mathbb{G}};T)$$. Remark 3.4 From equation (3.4) and by taking into account equations (2.9) and (2.10) we note that the optimal investment strategy depends on the information drift $$m$$ and the stochastic factor $$y$$. This is something very natural since: (i) we have assumed that the agent has an a priori knowledge of the value of the random variable $$\psi$$ and $$m$$ is the information drift induced by this future information. This of course means that the agent as an informed trader is taking advantage of her additional knowledge, as expected (ii) we have assumed that the stock prices are influenced by the external factor $$y$$. Hence, as also expected, the optimal investment strategy should take into account this dependence. Remark 3.5 From equation (3.4) we can see that the optimal investment policy consists of three terms, namely: \begin{align*} \pi^{*;0}(t,x,y) &= -\frac{\mu(y)}{x\sigma^{2}(y)}\frac{\hat{u}_{x}}{\hat{u}_{xx}}\\ \pi^{*;1}(t,x,y,m) &=\left[-m\frac{\hat{u}_{x}}{\hat{u}_{xx}} -f(t)\frac{\hat{u}_{xm}}{\hat{u}_{xx}}\right]\frac{1}{x\sigma(y)}\\ \pi^{*;2}(t,x,y) &=-\rho\frac{\beta(y)}{x\sigma(y)}\frac{\hat{u}_{xy}}{\hat{u}_{xx}}. \end{align*} The first component, $$\pi^{*;0}(t,x,y)$$, is known as the Merton investment strategy. This myopic behaviour corresponds to the investment policy followed by the economic agent under symmetric information in markets in which the investment opportunity set remains constant through time.7 Under the basis of our assumptions ($$x, \hat{u}_{x}>0$$ and $$\hat{u}_{xx}<0$$), the Merton portfolio is always positive for a non-zero market price of risk ($$\mu(y)>0$$). The second component, $$\pi^{*;1}(t,x,y,m)$$, is called the information signal hedging demand. It represents the additional investment caused by the knowledge of the information signal $$\psi$$. To be more precise, $$\pi^{*;1}(t,x,y,m)$$ behaves like a correction term to the Merton portfolio $$\pi^{*;0}(t,x,y)$$ so as to take into account the effect of the additional information. This term does not have a constant sign, for the signs of the mixed derivative $$\hat{u}_{xm}$$ and the information variable $$m$$ are not definite. Note that, if the information signal observed is completely imprecise, that is, it is only consisted of noise, this term should vanish (cf. Remark 4.4). The third component, $$\pi^{*;2}(t,x,y)$$, is called the excess hedging demand. It represents the additional investment caused by the presence of the stochastic factor. This term also does not have a constant sign, for the signs of the mixed derivative $$\hat{u}_{xy}$$ and the correlation coefficient $$\rho$$ are not definite. Note that the excess hedging demand vanishes in the uncorrelated case ($$\rho=0$$) and also when the volatility of the stochastic factor process is zero ($$\beta(y)=0$$), something however that is not allowed in our case. As already mentioned above (cf. Remark 4.4), when the inside information is not taken into account, the component $$\pi^{*;1}(t,x,y,m)$$ vanishes. In this case, the private information does not have anything to offer hence it is ignored; the manager operates under the filtration $$\mathbb{F}$$ and the optimal investment strategy is given by \begin{equation*} \pi^{*}(t,x,y) = \pi^{*;0}(t,x,y) + \pi^{*;2}(t,x,y). \end{equation*} 4. The case of the exponential utility function In the present section, in order to demonstrate the effectiveness of the proposed approach, we provide solutions in feedback form for the optimal investment strategy and the optimal value function, for the special case of the exponential utility function. In other words, we assume that the informed economic agent has exponential preferences, that is, a utility function of the form \begin{equation} U(x) = -\frac{1}{\gamma}e^{-\gamma x}, \end{equation} (4.1) where $$\gamma>0$$. This function has a constant absolute risk aversion parameter $$\gamma$$8 and plays an important role in both finance and insurance. Theorem 4.1 Assume exponential preferences (equation (4.1)). Then, the value function for the informed economic agent which is also the solution of the stochastic optimal control problem (3.1), admits the form \begin{equation} u(t,x,y,m)=-\frac{1}{\gamma}\exp\left[-\gamma xe^{r(T-t)} + g(t,y,m)\right], \end{equation} (4.2) where the function $$g$$ satisfies the following non-linear partial differential equation \begin{equation} \begin{aligned} g_{t} &+ \left[\alpha(y) - \rho\beta(y)\frac{\mu(y)}{\sigma(y)}\right]g_{y} - \frac{1}{2}\left[\frac{\mu(y)}{\sigma(y)} + m\right]^{2} +\frac{1}{2}\beta^{2}(y)\Big[(1-\rho^{2})g_{y}^{2}+g_{yy}\Big] \\ &-f(t)\left[\frac{\mu(y)}{\sigma(y)} + m\right]g_{m}+ \rho\beta(y) f(t)g_{ym}+\frac{1}{2}f^{2}(t)g_{mm} =0. \end{aligned} \end{equation} (4.3) with boundary condition $$g(T,y,m)=0$$. In this case, the optimal investment strategy is given in feedback form by \begin{equation} \pi^{*}(t,x,y,m)=\left[\left(\frac{\mu(y)}{\sigma(y)} + m\right) + \rho\beta(y)g_{y}+f(t) g_{m}\right]\frac{e^{-r(T-t)}}{\gamma x\sigma(y)}. \end{equation} (4.4) Proof. Suppose the partial differential equation (3.3) admits a classical solution $$u\in \mathcal{C}^{1,2,2,2}(\mathbb{S})$$ for every quadruple $$(t,x,y,m)\in\mathbb{S}$$. We look for solutions using the following ansatz: \begin{equation*} u(t,x,y,m) = -\frac{1}{\gamma}\exp\left\{ -\gamma x e^{r(T-t)} +g(t,y,m)\right\}\!, \end{equation*} where $$g(t,y,m)$$ is a suitable function with boundary conditions $$g(T,y,m)=0$$ (this follows from the boundary condition $$u(T,x,y,m)=U(x)$$), which will be determined later. Differentiating this trial solution with respect to $$(t,x,y,m)$$, yields \[ \begin{array}{ll} u_{t}(t,x,y,m)=u(t,x,y,m)\left[r\gamma xe^{r(T-t)} + g^{'}(t) \right] & u_{xm}(t,x,y,m)=u(t,x,y,m)\left[-\gamma e^{r(T-t)}\right]g_{m}\\ u_{x}(t,x,y,m)=u(t,x,y,m)\left[-\gamma e^{r(T-t)}\right] & u_{xy}(t,x,y,m)=u(t,x,y,m)\left[-\gamma e^{r(T-t)}\right]g_{y} \\ u_{m}(t,x,y,m) =u(t,x,y,m)g_{m} & u_{mm}(t,x,y,m)=u(t,x,y,m)\Big[g_{m}^{2} + g_{mm}\Big]\\ u_{y}(t,x,y,m) =u(t,x,y,m)g_{y} & u_{yy}(t,x,y,m)=u(t,x,y,m)\Big[g_{y}^{2} + g_{yy}\Big]\\ u_{xx}(t,x,y,m)=u(t,x,y,m)\left[\gamma e^{r(T-t)}\right]^{2} & u_{ym}(t,x,y,m)=u(t,x,y,m)\Big[g_{y}g_{m} + g_{ym}\Big]. \end{array}\] Substituting the above expressions in equation (3.3) leads to the non-linear partial differential equation (4.3) and in equation (3.4) gives the optimal control law (4.4) □ Corollary 4.1 In the complete market case $$(\rho=1)$$, the non-linear partial differential equation (4.3), reduces to the linear parabolic partial differential equation \begin{equation} \begin{aligned} g_{t} &+ \left[\alpha(y) -\beta(y)\frac{\mu(y)}{\sigma(y)}\right]g_{y} - \frac{1}{2}\left[\frac{\mu(y)}{\sigma(y)} + m\right]^{2} +\frac{1}{2}\beta^{2}(y)g_{yy}-f(t)\left[\frac{\mu(y)}{\sigma(y)} + m\right]g_{m}\\ &+ \beta(y) f(t)g_{ym}+ \frac{1}{2}f^{2}(t)g_{mm}=0, \end{aligned} \end{equation} (4.5) and in this case, the optimal investment decision is given by \begin{equation} \pi^{*}(t,x,y,m)=\left[\left(\frac{\mu(y)}{\sigma(y)} + m\right) + \beta(y)g_{y}+f(t) g_{m}\right]\frac{e^{-r(T-t)}}{\gamma x\sigma(y)}. \end{equation} (4.6) 4.1. A fully worked example In Theorem 4.1, we characterized the value function of the problem (3.1) as the solution of a non-linear partial differential equation and moreover derived the optimal investment policy, in the case of the exponential utility function (4.1). However, no further conclusions can be extracted since the form of the functions ($$\mu,\alpha,\beta$$ and $$\sigma$$) of the underlying driving system and the information signal $$\psi$$ are not explicitly defined. In the present section, for the sake of a more comprehensive analysis, we assume a specific form for the above parameters and present a fully worked example. To be more precise, in order to keep things as simple as possible, we adopt the complete market framework $$(\rho=1)$$, and assume that the parameters for the risky asset (2.2) and the stochastic factor process (2.3), are defined as \begin{align*} \mu(y) &= \mu_{0}\left(|y|+\delta\right)\\ \sigma(y) &= \sigma_{0}\left(|y|+\delta\right)\\ \alpha(y) &= \alpha_{0}(\theta-y)\\ \beta(y) &= \beta_{0}, \end{align*} where $$\mu_{0},\sigma_{0},\alpha_{0},\beta_{0}>0$$ and $$\theta\in\mathbb{R}$$. The parameter $$\delta$$ is a very small constant needed in order for Assumption 2.1 to be satisfied. In other words, the dynamics of the risky asset evolve according to the stochastic differential equation \begin{equation} \begin{aligned} \frac{dS(t)}{S(t)}&= \Big[r + \mu_{0}\Big(|Y(t)|+\delta\Big)\Big]dt + \sigma_{0}\Big(|Y(t)|+\delta\Big) dW(t)\\ S(0)&=S_{0}>0, \end{aligned} \end{equation} (4.7) while the stochastic factor process is driven by the Ornstein–Uhlenbeck process \begin{equation} \begin{aligned} dY(t)&= \alpha_{0}\left(\theta-Y(t)\right)dt + \beta_{0} dW(t)\\ Y(0)&=Y_{0} \in\mathbb{R}. \end{aligned} \end{equation} (4.8) The choice for the functions $$\mu,\sigma,\alpha$$ and $$\beta$$ for this example was made so that a realistic model which is analytically tractable within the reasonable page limits of this publication can be presented. Of course, more general options for the functions $$\mu,\sigma,\alpha$$ and $$\beta$$ can be used, allowing for more complex models and scenarios, that can be accommodated within Theorem 3.1. These are left to the interested reader. In what follows in order to avoid cluttering of notation we will use the notation $$\mu$$, $$\sigma$$, $$\alpha$$, $$\beta$$ for the constants $$\mu_0$$, $$\sigma_0$$, $$\alpha_0$$ and $$\beta_0$$. Remark 4.1 The above specific choice for the market parameters is mainly driven by the requirements of Assumption 2.1 together with the fact that an Ornstein–Uhlenbeck type process is a very popular choice for the stochastic factor (e.g. interest rate or stochastic volatility), see for example Bielecki & Pliska (2001), Delong & Klüppelberg (2008) or Fouque et al. (2000) and references therein. However, note that the requirements of Assumption 2.1, especially the globally Lipschitz assumption, exclude some well-known models such as the Heston or the Hull and White stochastic volatility models. At this point, take into account that the model described by equations (4.7) and (4.8), is not aimed to serve as the most realistic model (if there exists one) but to give an insight on how the proposed framework works in practice, keeping of course in line with Assumption 2.1. This model will not only provide explicit closed form solutions for the value function and the optimal investment strategy but also lay the groundwork for a detailed numerical analysis in Section 5. Additionally, we consider $$T_{0}=T$$ within the setting described in Section 2.3, and assume that the information signal $$\psi$$ admits the form \begin{equation} \psi = \lambda W(T) + (1-\lambda)\epsilon, \end{equation} (4.9) where $$\epsilon$$ is a standard normal random variable, independent of $$W$$. In this case, the manager possesses information about the final value of the Brownian motion $$W(t)$$ in the trading interval $$[0,T]$$, which is distorted by some observation noise $$\epsilon$$. A value of $$\lambda=1$$ means that the manager explicitly knows the value of $$W(T)$$, whereas, a value of $$\lambda=0$$ means that the manager observes nothing but noise. In this setting, we are only interested in non-trivial case $$0\leq\lambda<1$$, that is, the available information comes with noise. For the case at hand, it is well known (see e.g. Fei & Wu, 2003 or Pikovsky & Karatzas, 1996), that the information drift admits the form \begin{equation} m(t) = \frac{\big[\psi - \lambda W(t)\big]\lambda}{\lambda^{2}(T-t) + (1-\lambda)^{2}}, \end{equation} (4.10) for every $$t\in[0,T]$$ and $$0\leq\lambda<1$$. In fact, equation (4.10) is fully compatible with our Assumptions regarding the information drift. To be more precise, concerning Assumption 2.2, we have the following well-known result. Remark 4.2 (Pikovsky & Karatzas, 1996) The information drift (4.10), satisfies \begin{equation*} \mathbb{E}\left[\int_{0}^{T}m^{2}(s)ds\right]\leq\frac{C}{(1-\lambda)^{2}}<\infty, \end{equation*} for $$\lambda<1$$ and some constant $$C>0$$. Remark 4.3 In the adopted framework, the portfolio manager observes, from the beginning of the trading interval, a signal $$\psi$$ concerning the terminal value of the underlying Brownian motion $$W$$. However, this signal is not clear, in the sense that it is subjected to some observation noise $$\epsilon$$. As a result, for any value of $$\lambda\in[0,1)$$, the final value $$W(T)$$ is not precisely known. This fact prevents us from resorting to the classical theory of Brownian bridge enlargements. Concerning the justification of Assumption 2.3, we have the next result. Proposition 4.1 For the information drift defined by equation (4.10), we have that \begin{align} f(t) = -\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}} \end{align} (4.11) and \begin{align} \hspace{-2.8em}m(0)= \frac{\lambda \psi}{\lambda^{2}T + (1-\lambda)^{2}}. \end{align} (4.12) Proof. First, note that according to equation (2.9), under the enlarged observed filtration $$\mathbb{G}$$, the process $$W(t)$$ is an Itô process with decomposition given by \begin{equation*} dW(t) = d\widetilde{W}(t) + m(t)dt, \end{equation*} where $$\widetilde{W}(t)$$ is a $$\mathbb{G}$$-Brownian motion. A straightforward application of Itô’s lemma to the $$\mathcal{C}^{1,2}\left([0,T]\times\mathbb{R}\right)$$ function $$m(t,W(t))$$, yields (observe equation (4.10)) \begin{align*} dm(t,W(t)) &= \frac{\partial m(t,W(t))}{\partial t}dt + \frac{\partial m(t,x)}{\partial x}\Big|_{x=W(t)}dW(t) + \frac{1}{2}\frac{\partial^{2} m(t,x)}{\partial x^{2}}\Big|_{x=W(t)}(dW(t))^{2} \nonumber \\ &= \frac{\lambda^{3}\left[\psi - \lambda W(t)\right]}{\left[\lambda^{2}(T-t) + (1-\lambda)^{2}\right]^{2}}dt -\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\Bigg[d\widetilde{W}(t) + \frac{\big[\psi - \lambda W(t)\big]\lambda}{\lambda^{2}(T-t) + (1-\lambda)^{2}}dt \Bigg] \nonumber \\ &=-\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}d\widetilde{W}(t), \end{align*} and equation (4.11) follows by comparing the above equation, with equation (2.10). The proof is concluded by setting in equation (4.10), $$t=0$$ and taking into account that $$W$$ is assumed to be a standard Brownian motion ($$W(0)=0$$). □ Theorem 4.2 Assume exponential preferences (equation (4.1)) and moreover that the risky asset evolves according to the stochastic differential equations (4.7) and (4.8) and that the information signal is given by equation (4.9). The value function for the informed economic agent which is also the solution of the stochastic optimal control problem (3.1), admits the form \begin{equation} u(t,x,y,m)=-\frac{1}{\gamma}\exp\left[-\gamma xe^{r(T-t)} + g(t,y,m)\right], \end{equation} (4.13) where \begin{equation} g(t,y,m) = A_{1}(t)m^{2} + A_{2}(t)m + A_{3}(t)y^{2} + A_{4}(t)y+ A_{5}(t)my + A_{6}(t), \end{equation} (4.14) with \begin{align} A_{1}(t) &= -\frac{T-t}{2(1-\lambda)^{2}}\Big[\lambda^{2}(T-t) + (1-\lambda)^{2}\Big]\\ \end{align} (4.15) \begin{align} A_{2}(t) &= -\frac{\mu}{\sigma}\frac{T-t}{(1-\lambda)^{2}}\Big[\lambda^{2}(T-t) + (1-\lambda)^{2}\Big]\\ \end{align} (4.16) \begin{align} A_{3}(t) &= A_{4}(t)= A_{5}(t) = 0\\ \end{align} (4.17) \begin{align} A_{6}(t) &= \frac{1}{2(1-\lambda)^{2}}\left\{-\lambda^{2}(T-t)+ (1-\lambda)^{2}\log\left[\frac{(1-\lambda)^{2} + \lambda^{2}(T-t)}{(1-\lambda)^{2}}\right]\right\} \\ &\quad-\frac{1}{2}\left(\frac{\mu}{\sigma}\right)^{2}(T-t) \left[1 + \frac{\lambda^{2}}{(1-\lambda)^{2}}(T-t)\right].\nonumber \end{align} (4.18) In this case, the optimal investment strategy for the informed economic agent is given by the feedback rule \begin{equation} \pi^{*}(t,x,y,m)=\frac{\mu + \sigma m}{\sigma}\frac{\lambda^{2}(T-t) + (1-\lambda)^{2}}{(1-\lambda)^{2}}\frac{e^{-r(T-t)}}{\gamma x\sigma(|y|+\delta)}. \end{equation} (4.19) Proof. For the case at hand (note that $$f(t)$$ is given by equation (4.11)), equation (4.5) leads to \begin{equation} \begin{aligned} g_{t} &+ \left[\alpha(\theta-y) - \beta\frac{\mu}{\sigma}\right]g_{y} - \frac{1}{2}\left(\frac{\mu}{\sigma} + m\right)^{2} +\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\left(\frac{\mu}{\sigma}+m\right)g_{m}\\ &+\frac{1}{2}\left[\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\right]^{2}g_{mm}-\frac{\beta\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}g_{ym} +\frac{1}{2}\beta^{2}g_{yy} =0. \end{aligned} \end{equation} (4.20) We conjecture a solution to the linear parabolic partial differential equation (4.20), with the following form \begin{equation*} g(t,y,m) = A_{1}(t)m^{2} + A_{2}(t)m + A_{3}(t)y^{2} + A_{4}(t)y+ A_{5}(t)my + A_{6}(t), \end{equation*} where $$A_{1}(t),A_{2}(t),A_{3}(t),A_{4}(t),A_{5}(t)$$ and $$A_{6}(t)$$ are suitable functions to be determined later with boundary condition $$A_{1}(T)=A_{2}(T)=A_{3}(T)=A_{4}(T)=A_{5}(T)=A_{6}(T)=0$$ (this follows from the boundary condition $$g(T,y,m)=0$$). Substituting this trial solution in equation (4.20) we get the following ordinary differential equations \begin{align} &A_{1}'(t) + \frac{2\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}A_{1}(t) - \frac{1}{2}=0 \\ \end{align} (4.21a) \begin{align} &A_{2}'(t) + \frac{A_{2}(t)\lambda^{2}}{\lambda^{2}(T-t)+(1-\lambda)^{2}} + \frac{2\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\frac{\mu}{\sigma}A_{1}(t)+\left(\alpha\theta-\beta\frac{\mu}{\sigma}\right)A_{5}(t) - \frac{\mu}{\sigma}=0 \\ \end{align} (4.21b) \begin{align} &A_{3}'(t) - 2\alpha A_{3}(t)=0 \\ \end{align} (4.21c) \begin{align} &A_{4}'(t) - \alpha A_{4}(t) + 2\left(\alpha\theta - \beta\frac{\mu}{\sigma}\right)A_{3}(t) + \frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\frac{\mu}{\sigma}A_{5}(t) =0 \\ \end{align} (4.21d) \begin{align} & A_{5}'(t) - \Bigg[a - \frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\Bigg]A_{5}(t) = 0 \\ \end{align} (4.21e) \begin{align} & A_{6}'(t) + \left(\alpha\theta - \beta\frac{\mu}{\sigma}\right)A_{4}(t) -\frac{1}{2}\left(\frac{\mu}{\sigma}\right)^{2} +\beta^{2}A_{3}(t) + \left[\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\right]^{2}A_{1}(t) \nonumber \\ &\qquad +\frac{\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}\frac{\mu}{\sigma}A_{2}(t) - \frac{\beta\lambda^{2}}{\lambda^{2}(T-t) + (1-\lambda)^{2}}A_{5}(t) =0. \end{align} (4.21f) In order to solve the above ordinary differential equations, we proceed as follows: (i) we solve equations (4.21a), (4.21c) and (4.21e), (ii) we substitute the solutions we found for equations (4.21a) and (4.21e) into equation (4.21b) and then solve it, (iii) we substitute the solutions we found for equations (4.21c) and (4.21e) into equation (4.21d) and then solve it (iv) and finally, substitute the solutions for equations (4.21a–4.21e) into equation (4.21f) and then solve it. This eventually yields to equation (4.13) which is the optimal value function associated with the stochastic optimal control problem at hand. Regarding the optimal investment strategy, from equation (4.6), adapted to our specific example, we observe that \begin{align*} &\pi^{*}(t,x,y,m)\\ &\quad=\left[\left(\frac{\mu(y)}{\sigma(y)} + m\right) + \beta(y)g_{y}+f(t) g_{m}\right]\frac{e^{-r(T-t)}}{\gamma x\sigma(y)} \\ &\quad=\left[\frac{\mu + \sigma m}{\sigma} + \beta\left(2A_{3}(t)y + A_{4}(t) + A_{5}(t)m\right) - \frac{\lambda^{2}\left(2A_{1}(t)m + A_{2}(t) + A_{5}(t)y\right)}{\lambda^{2}(T-t)+(1-\lambda)^{2}}\right]\frac{e^{-r(T-t)}}{\gamma x\sigma(|y|+\delta)} \\ &\quad=\left[\frac{\mu + \sigma m}{\sigma} - \frac{\lambda^{2}}{\lambda^{2}(T-t)+(1-\lambda)^{2}}\left(2A_{1}(t)m + A_{2}(t)\right)\right]\frac{e^{-r(T-t)}}{\gamma x\sigma(|y|+\delta)} \\ &\quad=\frac{\mu + \sigma m}{\sigma}\frac{\lambda^{2}(T-t) + (1-\lambda)^{2}}{(1-\lambda)^{2}}\frac{e^{-r(T-t)}}{\gamma x\sigma(|y|+\delta)}, \end{align*} which follows from equation (4.13) and the solution of the ordinary differential equations (4.21a)–(4.21f). This completes the proof □ Remark 4.4 Regarding the optimal investment strategy, by observing equation (4.19), we distinguish the following two interesting cases concerning the possible values of $$\lambda$$: $$\underline{\text{When}\,{\lambda=0}}$$, the information signal becomes $$\psi=\epsilon$$, that is, it is only consisted of noise. In this case, the optimal investment strategy is given in feedback form by \begin{equation*} \pi^{*}(t,x,y) =\frac{\mu}{\sigma}\frac{e^{-r(T-t)}}{\gamma x\sigma(|y|+\delta)}, \end{equation*} (and is obtained by setting in equations (4.10) and (4.19), $$\lambda=0$$). This is the optimal Markovian control law associated with the stochastic optimal control problem \begin{equation*} u(t,x,y) = \sup_{\pi\in\mathcal{A}({\mathbb{F}};T)} \mathbb{E}\Big[U\left(X^{\pi}(T)\right) \big| \mathscr{F}_{t}\Big], \end{equation*} subject to the state process \begin{equation*} \begin{aligned} \frac{dX^{\pi}(t)}{X^{\pi}(t)} &= \Big[r + \pi(t)\mu\left(|Y(t)|+\delta\right)\Big]dt + \pi(t)\sigma\left(|Y(t)|+\delta\right)dW(t) \\ dY(t) &=\alpha\left(\theta-Y(t)\right)dt + \beta dW(t), \end{aligned} \end{equation*} since in this case $$m(t)=0$$ for every $$t\in[0,T]$$. Stated differently, when $$\lambda=0$$ the economic agent ignores the information signal, as it has nothing special to offer, and operates under the publicly available information flow $$\mathbb{F}$$. $$\underline{\text{When}\,{\lambda=1}}$$, the information signal becomes $$\psi=W(T)$$, that is, the final value of the underlying Brownian motion driving uncertainty in the stock price process (4.7), is explicitly available at the agent’s disposal from the very beginning of the trading interval. In this case, \begin{equation} m(t) \longrightarrow \frac{W(T)-W(t)}{T-t}. \end{equation} (4.22) As $$t\rightarrow T^{-}$$ the right hand side in equation (4.22) behaves like the non-existing derivative of the Brownian motion $$\left\{W(t),\, t\geq 0\right\}$$ at the point $$t=T$$, hence the aforementioned strategy is impossible. Such a strategy would be characterized by wild fluctuations around $$T$$ and this behaviour might act as an indicator the economic agent operates under the influence of private information. Moreover, it is well known from Pikovsky & Karatzas (1996), that in this case the value of the problem becomes infinite and arbitrage opportunities arise. For more information on this subject, we refer the interested reader to the article of Liu & Longstaff (2004). Remark 4.5 If we set in equation (4.9), $$\lambda=1$$ and consider the future time $$T_{0}>T$$, we retrieve the case $$\psi = W(T_{0})$$, where $$T_{0}>T$$. That is, the manager explicitly knows the value of the underlying Brownian motion $$W(t)$$ at the future time $$T_{0}$$. In this case, it is well-known (see e.g. Pikovsky & Karatzas, 1996; Mansuy & Yor, 2006) that the information drift is given by \begin{equation} m(t) = \frac{\psi - W(t)}{T_{0}-t}. \end{equation} (4.23) Furthermore, Assumption 2.2 is satisfied for any $$T<T_{0}$$, and by an application of Itô’s lemma to the $$\mathcal{C}^{1,2}\left([0,T]\times\mathbb{R}\right)$$ function $$m(t,W(t))$$, defined in equation (4.23), we get that \begin{equation} f(t) = -\frac{1}{T_{0}-t} \end{equation} (4.24) and \begin{equation} m(0) = \frac{\psi}{T_{0}}. \end{equation} (4.25) As a result, we can see that Assumption 2.3 is again justified. In order to have a complete picture for the problem at hand, we provide a result similar to Theorem 4.2, adapted to the setting of Remark 4.5. Theorem 4.3 Assume exponential preferences (equation (4.1)) and moreover that the risky asset evolves according to the stochastic differential equations (4.7) and (4.8) and that the information signal is given $$\psi=W(T_{0}), T_{0}>T$$. The value function for the informed economic agent which is also the solution of the stochastic optimal control problem (3.1), admits the form \begin{equation} u(t,x,y,m)=-\frac{1}{\gamma}\exp\left[-\gamma xe^{r(T-t)} + \zeta(t,y,m)\right], \end{equation} (4.26) where \begin{equation} \zeta(t,y,m) = F_{1}(t)m^{2} + F_{2}(t)m + F_{3}(t)y^{2} + F_{4} (t)y+ F_{5}(t)my + F_{6}(t), \end{equation} (4.27) with \begin{align} F_{1}(t)&=-\frac{(T_{0}-t)(T-t)}{2(T_{0}-T)} \\[3pt] \end{align} (4.28) \begin{align} F_{2}(t)&=-\frac{\mu(T_{0}-t)(T-t)}{\sigma(T_{0}-T)} \\[3pt] \end{align} (4.29) \begin{align} F_{3}(t)&=F_{4}(t)=F_{5}(t)=0\\[3pt] \end{align} (4.30) \begin{align} F_{6}(t)&=-\left(\frac{\mu}{\sigma}\right)^{2}\frac{(T_{0}-t)(T-t)}{2(T_{0}-T)} + \frac{1}{2}\Bigg[ \log\left[\frac{T_{0}-t}{T_{0}-T}\right] -\frac{T-t}{T_{0}-T} \Bigg]. \end{align} (4.31) In this case, the optimal investment strategy for the informed agent is given in feedback form by \begin{equation} \pi^{*}(t,x,y,m) = \frac{\mu + \sigma m}{\sigma}\frac{T_{0}-t}{T_{0}-T}\frac{e^{-r(T-t)}}{\gamma x \sigma (|y|+\delta)}. \end{equation} (4.32) Proof. For the case at hand (note that $$f(t)$$ is given by (4.24)), equation (4.5) leads to \begin{equation} \begin{aligned} \zeta_{t} &+ \left[\alpha(\theta-y) - \beta\frac{\mu}{\sigma}\right]\zeta_{y} - \frac{1}{2}\left(\frac{\mu}{\sigma} + m\right)^{2} + \frac{1}{T_{0}-t}\left(\frac{\mu}{\sigma}+m\right)\zeta_{m}\\ &+\frac{1}{2}\left(\frac{1}{T_{0}-t}\right)^{2}\zeta_{mm}-\frac{\beta}{T_{0}-t}\zeta_{ym} +\frac{1}{2}\beta^{2}\zeta_{yy} =0. \end{aligned} \end{equation} (4.33) We conjecture a solution to the linear parabolic partial differential equation (4.33), with the following form \begin{equation*} \zeta(t,y,m) = F_{1}(t)m^{2} + F_{2}(t)m + F_{3}(t)y^{2} + F_{4}(t)y+ F_{5}(t)my + F_{6}(t), \end{equation*} where $$F_{1}(t),F_{2}(t),F_{3}(t),F_{4}(t),F_{5}(t)$$ and $$F_{6}(t)$$ are suitable functions to be determined later with boundary condition $$F_{1}(T)=F_{2}(T)=F_{3}(T)=F_{4}(T)=F_{5}(T)=F_{6}(T)=0$$ (this follows from the boundary condition $$\zeta(T,y,m)=0$$). Substituting this trial solution in equation (4.33), we get the following ordinary differential equations \begin{align} &F_{1}'(t) + \frac{2F_{1}(t)}{T_{0}-t} - \frac{1}{2}=0 \\ \end{align} (4.34a) \begin{align} &F_{2}'(t) + \frac{F_{2}(t)}{T_{0}-t} + \frac{2}{T_{0}-t}\frac{\mu}{\sigma}F_{1}(t)+\left(\alpha\theta-\beta\frac{\mu}{\sigma}\right)F_{5}(t) - \frac{\mu}{\sigma}=0 \\ \end{align} (4.34b) \begin{align} &F_{3}'(t) - 2\alpha F_{3}(t)=0 \\ \end{align} (4.34c) \begin{align} &F_{4}'(t) - \alpha F_{4}(t) + 2\left(\alpha\theta - \beta\frac{\mu}{\sigma}\right)F_{3}(t) + \frac{1}{T_{0}-t}\frac{\mu}{\sigma}F_{5}(t) =0 \\ \end{align} (4.34d) \begin{align} & F_{5}'(t) - \Bigg[a - \frac{1}{T_{0}-t}\Bigg]F_{5}(t) = 0 \\ \end{align} (4.34e) \begin{align} & F_{6}'(t) + \left(\alpha\theta - \beta\frac{\mu}{\sigma}\right)F_{4}(t) -\frac{1}{2}\left(\frac{\mu}{\sigma}\right)^{2} +\beta^{2}F_{3}(t) + \left[\frac{1}{T_{0}-t}\right]^{2}F_{1}(t) \nonumber \\ &\qquad +\frac{1}{T_{0}-t}\frac{\mu}{\sigma}F_{2}(t) - \frac{\beta}{T_{0}-t}F_{5}(t) =0. \end{align} (4.34f) Solving the above system of ordinary differential equations (by following the same approach as in the proof of Theorem 4.2), leads to the optimal value function (4.26) and to the optimal investment strategy (4.32) □ 5. Numerical study of the optimal investment strategy According to the concrete example of Section 4.1, the portfolio manager, from the very beginning of the trading interval, has exclusive access to some additional information which stems from the observation of a private signal $$\psi$$ concerning the value of $$W(T)$$, as it is defined in equation (4.9). However, this signal is not precise, in the sense that it is subject to some normally distributed observation noise $$\epsilon$$. Having the value of $$\psi$$ in mind, the investor, tries to make the best possible estimation about the value of $$Y(T)$$ and in advance about the value of $$S(T)$$. Based on this prediction, she then decides the optimal investment policy to be followed, which in the special case: the financial market is driven by the system of stochastic differential equations (4.7) and (4.8) the information signal admits the form (4.9) and under exponential preferences (equation (4.1)) is given in feedback form by \begin{equation} \pi^{*}(t,X^{*}(t),Y(t),m(t))=\frac{\mu + \sigma m(t)}{\sigma}\frac{\lambda^{2}(T-t) + (1-\lambda)^{2}}{(1-\lambda)^{2}}\frac{e^{-r(T-t)}}{\gamma X^{*}(t)\sigma\left(|Y(t)|+\delta\right)}, \end{equation} (5.1) where $$X^{*}(t)=X^{\pi^{*}}(t)$$ denotes the wealth process under the optimal investment strategy (the dynamics of $$X^{*}(t)$$ can be easily obtained by substituting equation (5.1) in the SDE that describes the wealth of the economic agent, see equation (2.11)). Hence, as far as the simulation of the optimal investment strategy is concerned, special treatment is needed in order to effectively capture the behaviour of the informed agent, with the major obstacle being the simulation of $$X^{*}(t), Y(t)$$ and $$m(t)$$. In this endeavour, we follow the next steps: (i) Numerically solve the stochastic differential equations for $$X^{*}(t), Y(t)$$ and $$m(t)$$ by employing an Euler–Maruyama scheme. In order to implement the method, for a time step of size $$\Delta t=T/N$$ with $$N=2^{11}$$ points, we define the step size in the Euler–Maruyama scheme as $$\delta t=\Delta t$$. (ii) Based on step (i), calculate the mean over a large number of realizations of the optimal investment strategy $$\pi^{*}$$, as defined in equation (5.1). An appropriate method to attack this problem is the Monte Carlo simulation. In order to implement the Monte Carlo method, we simulate a large number M of paths of $$\pi^{*}$$ in the time interval $$[0,T]$$ and at each time point we plot the average of M different values. We also use for each path $$N = 2^{\alpha}$$ number of points (here $$N=2^{11}$$ and $$M=6000$$ paths). In what follows, unless stated otherwise, we let $$\Pi=[0,1]$$, $$T=1$$ year, $$X(0)=1.5$$ and $$\gamma=1.5$$. The parameters of the financial market are chosen as $$\mu=10\%$$, $$r=6\%$$, $$\sigma=40\%$$ and $$\delta=10^{-3}$$. The parameters for the external factor are chosen as $$Y(0)=1.5$$, $$\alpha=2$$, $$\theta=1$$ and $$\beta=40\%$$. Remark 5.1 When employing an Euler–Maruyama scheme, one is usually interested in the convergence of the scheme, that is, if the simulated solution converges to the exact solution of the stochastic differential equation at hand. Typically, the parameters must satisfy some globally Lipschitz and a non-explosion condition (see e.g. Gilsing & Shardlow, 2007). In our case, convergence seems to hold, at least in a weak sense. However, we do not examine this matter any further, since it exceeds the scope of the current work. 5.1. Simulation results From Fig. 1, we have some very interesting findings: Fig. 1. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths. In this case we consider $$\lambda=0.2$$. (b) Average path of 6000 optimal investment strategy paths for various values of the parameter $$\lambda$$. In this case we consider $$\psi=0.2$$. Fig. 1. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths. In this case we consider $$\lambda=0.2$$. (b) Average path of 6000 optimal investment strategy paths for various values of the parameter $$\lambda$$. In this case we consider $$\psi=0.2$$. From Fig. 1(a), we can see that as the level of the value of $$\psi$$ increases, the economic agent is expected to invest more in the stock market compared to the symmetric information case. This seems quite natural, since an increasing level for the information signal leads to an increasing value for $$W(T)$$ which in advance leads to an optimistic prediction for $$S(T)$$. With this prediction at hand, the economic agent expects higher returns for the underlying stock leading to more interest in the risky market. On the other hand, a decreasing value for $$\psi$$ leads to pessimistic predictions for $$S(T)$$ and as a result the economic agent turns her attention to the risk-free asset in an attempt to seek safer investment opportunities. As the parameter $$\lambda$$ increases, the investor is expected to follow what the private information dictates. That is, she is expected to invest more in the risky asset (as the level of the information signal increases) and to turn her attention in the bond market (as the level of the information signal decreases). On the other hand, as $$\lambda\rightarrow 0$$, as already stated in Remark 4.4, the information signal becomes very imprecise and has nothing useful to offer. In this case, from Fig. 1(b), we can see that the optimal investment strategy of the informed agent converges to the behaviour of the honestly informed agent. In fact, this result coincides with the findings of Remark 4.4. From Fig. 2(a,b), we have the following interesting results: From Fig. 2(a), we can see that as the risk aversion parameter $$\gamma$$ increases, the economic agent is expected to invest more in the risk-free asset. This also seems very natural, since in this case the agent is too risk averse to take advantage of the private information she is offered, and seeks safer investment opportunities in the risk-free market. In a similar fashion, from Fig. 2(b), we can see that as the initial wealth level of the economic agent increases, the agent is expected to ignore the private information and turn her attention in the risk-free asset. This behaviour, which in the current framework is a byproduct of the exponential utility function (4.1), is in contrast with the behaviour of the investor in Liu et al. (2010); In their setting, under power-type preferences, the more wealthy the agent the more valuable is the information. On the other hand, in our setting, the more wealthy the manager is, the less is interested in the additional information and the risky asset, and enjoys the risk-free interest rate. Fig. 2. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths for various values of the parameter $$\gamma$$. In this case we consider $$\psi=0.5$$ and $$\lambda=0.2$$. (b) Average path of 6000 optimal investment strategy paths for various values of the initial wealth. In this case we consider $$\psi=0.5$$ and $$\lambda=0.2$$. Fig. 2. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths for various values of the parameter $$\gamma$$. In this case we consider $$\psi=0.5$$ and $$\lambda=0.2$$. (b) Average path of 6000 optimal investment strategy paths for various values of the initial wealth. In this case we consider $$\psi=0.5$$ and $$\lambda=0.2$$. Finally, from Fig. 3(a,b), we have the following interesting results: Fig. 3. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths for various values of the initial level of the Ornstein–Uhlenbeck stochastic factor. In this case we consider $$\psi=0$$ and $$\lambda=0.3$$. (b) Average path of 6000 optimal investment strategy paths for various values of parameter $$\theta$$ of the Ornstein–Uhlenbeck stochastic factor. In this case we consider $$\psi=0.3$$ and $$\lambda=0.2$$. Fig. 3. View largeDownload slide (a) Average path of 6000 optimal investment strategy paths for various values of the initial level of the Ornstein–Uhlenbeck stochastic factor. In this case we consider $$\psi=0$$ and $$\lambda=0.3$$. (b) Average path of 6000 optimal investment strategy paths for various values of parameter $$\theta$$ of the Ornstein–Uhlenbeck stochastic factor. In this case we consider $$\psi=0.3$$ and $$\lambda=0.2$$. From Fig. 3(a), we can see that as the initial level of the stochastic factor process increases, the economic agent is expected to invest more in the risk-free asset. In fact, this is explained as follows. When the financial market evolves according to the stochastic differential equations (4.7) and (4.8), as the level of the stochastic factor increases, together with the market risk premium, the systematic risk associated with the risky asset (4.7) increases too, but the Sharpe ratio of the portfolio is always the same. In other words, the manager is not properly compensated for the additional risk she is willing to undertake. This makes the risky market (4.7) and (4.8) not a reasonable investment. Hence, investing in the risk-free asset seems the best alternative in this case. Additionally, we observe (by taking into account Fig. 1(a)) that even a value of $$\psi=0$$ leads to a more aggressive position in the risky market, compared to the symmetric information case. From Fig. 3(b), we can see that as the parameter $$\theta$$ increases, the economic agent turns her attention in the risk-free asset. In the current example, the stochastic factor evolves according to an Ornstein–Uhlenbeck process and $$\theta$$ is the long-term mean. A high value of the parameter $$\theta$$ means higher values for the stochastic factor, leading to more investment in the risk-free asset (according to the previous explanation). Remark 5.2 The effect of the volatility parameters $$\sigma$$ and $$\beta$$ is the same with the effect of $$\gamma$$, hence their numerical study is committed. 6. Conclusions In the present article, we studied an optimal control problem for a general stochastic factor model under the existence of some private information. To be more precise, we considered a portfolio manager, who has the possibility to invest part of her wealth in a financial market consisting of a risk-free asset and a risky one, whose coefficients depend on some exogenous correlated diffusion process, known as the stochastic factor. Additionally, the manager, from the very beginning of the trading interval, observes some information signal concerning the future. However, this signal is not precise in the sense it is subject to some observation noise. Within this very general framework, by resorting to the mixture of dynamic programming and initial enlargement of filtrations techniques, we effectively attacked an expected utility maximization problem by taking into account the enlarged information set of the manager. In this vein, we provided a general result characterizing the value function of the problem at hand and the optimal control law, by solving the associating Hamilton–Jacobi–Bellman equation. Moreover, in the special case, when: (i) the manager operates under exponential preferences, (ii) the stochastic factor process evolves according to an Ornstein–Uhlenbeck process and (iii) the information signal concerns a future noisy value of the Brownian motion implying uncertainty in the stock price process, we provided closed form solutions for the optimal investment decision and the optimal value function. In addition, by employing an Euler–Maruyama scheme followed by a Monte-Carlo approach, we numerically studied this special example, capturing in this way the qualitative and quantitative features of the private information on the optimal investment strategy. The model is extended in Supplementary Material in two major ways: (i) we solve the proposed model for other utility function, namely, in the case of the logarithmic and the power utilities and (ii) we present the explication of our framework in the multi-dimensional case, that is, when the market is consisted of multiple risky assets and multiple stochastic factors. Supplementary Material Supplementary Material is available at http://www.imaman.oxfordjournals.org/. Acknowledgements The authors would like to thank the editor and the three anonymous referees for their constructive comments that led to an improvement of this work. Part of this research was conducted while the first author was a Postdoctoral Research Fellow at the Department of Statistics of the Athens University of Economics and Business is gratefully acknowledged. Funding I.B. was supported by the research funding program: “Research Funding at Athens University of Economics and Business for Excellence and Extroversion: Action 2”: EP-2448-01/01-01, EP-2448-01/01-02, duration: 18/02/2016-31/01/2017. Footnotes 1Fei & Wu (2003) study an optimal investment problem under the existence of some private information in the case of a large investor whose actions affect the market prices. 2Which requires the conditional distribution of $$\psi$$ given $$\mathscr{F}_{t}$$ to be absolutely continuous to the law of $$\psi$$. 3For example, one could consider the specific case $$\psi = \lambda W(T_{0})+(1-\lambda)\epsilon$$, with $$T_{0}>T$$, where $$\epsilon$$ is a standard normal random variable which is independent of $$W$$. 4For a further investigation of the relationship between the additional information available and the information drift in a general setting, we refer the interested reader to the article of Amendinger & Imkeller (2007). 5These arguments require the function $$\hat{u}$$ to satisfy a quadratic growth condition. 6Under the additional assumption that the coefficients of the market are bounded by some large constant and by taking into account equation (2.2). 7A similar result can also be found in Zariphopoulou (2009), who studied an optimal investment problem within a stochastic factor model under symmetric information. 8This can be seen from the fact that $$-\frac{u^{''}(x)}{u^{'}(x)}=\gamma$$. 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Published: Dec 8, 2017

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