TY - JOUR
AU - Ma,, Dan
AB - Abstract The interest rate and the market price of risk may be stochastic in a real-world financial market. In this paper, the interest rate is assumed to be driven by a stochastic affine interest rate model and the market price of risk from the stock market is a mean-reverting process. In addition, the dynamics of the stock are simultaneously driven by random sources of interest rate and the stock market itself. In pension fund management, different fund managers may have different risk preferences. We suppose risk preference is described by the hyperbolic absolute risk aversion utility, which is a general utility function describing different risk preferences. Legendre transform-dual theory is presented to successfully obtain explicit expressions for optimal strategies. A numerical example illustrates the sensitivity of optimal strategies to market parameters. Theoretical results imply that the risks from stochastic interest rate and stochastic return may be completely hedged by adopting specific portfolios. 1. Introduction With the increasing rise of population and the coming of ageing population, the risk level stemmed from the pension fund system in many countries is increasing. Meanwhile, lower interest rates are also a big reason for greater risk, which gives investors few alternatives to achieve significant returns. More importantly, future liabilities are increased in a low interest rate regime. How to design an effective pension scheme has become the focus of concerns. The defined benefit (DB) plan and the defined contribution (DC) plan are two main types of plans in the design of pension plans. In DB plans, the benefits are defined in advance and the contributions are set and subsequently adjusted to keep in balance. In addition, the risks are shared by sponsors and employees, as shown in the paper of Haberman et al. (2003). However, in DC plans, the contributions are defined and the financial risks are shifted from sponsors to employees. Compared with DB plans, many sponsors find out that DC plans are more suitable to pension markets and have been adopted by many countries. In DC plans, the benefits after retirement depend on the investment returns at the accumulative phase; the optimal portfolios are crucial to the DC pension management. In recent years, some scholars investigated DC pension plans from the mean-variance point of view and achieved many important results. The interested readers may refer to the works of Yao et al. (2013), Guan & Liang (2015), Wu & Zeng (2015), Yao et al. (2016) and Li et al. (2016, 2017). Specifically, Yao et al. (2013) considered an optimal strategy for a scheme member facing the stochastic inflation and supposed all the assets in the financial market to be risky, where the instantaneous contribution of the pension changed according to the inflation level. Guan & Liang (2015) investigated the optimization problem for a DC plan in a stochastic affine interest rate model, where the return of risky asset was a mean-reverting process and the contribution rate was stochastic and was jointly affected by the dynamics of interest rate and stock index. Wu & Zeng (2015) studied a multiperiod time-inconsistent DC plan with stochastic salary flow and stochastic mortality risk in a generalized mean-variance framework. Yao et al. (2016) concerned a multiperiod mean-variance DC scheme with stochastic income and mortality risk in a Markovian regime-switching financial market. Li et al. (2016) studied a time-inconsistent DC pension plan with stochastic salary under the constant elasticity of variance (CEV) model, where the contribution rate is proportional to the members’ salary. Afterwards, Li et al. (2017) incorporated the default risk and return of premiums clauses into a DC pension plan with the CEV model and discussed the effect of return of premiums clauses and default mechanisms on the equilibrium strategies and fund size level. On DC pension plans in a mean-variance framework, readers can also refer to the works of Vigna (2014) and Sun et al. (2016) and so on. The above papers provided solid theoretical foundations for DC pension management problems in the mean-variance framework. It is well known that the utility criterion and the mean-variance criterion are two different tools in dealing with portfolio selection problems. Compared with the mean-variance criterion, the utility criterion can describe the risk preference of fund managers and can be solved by more methods, for example, Chiu & Wong (2018) and Zhao & Rong (2017). So it is widely used in the field of portfolio selection theory. Therefore, studying and solving DC pension fund problems in the expected utility maximization framework also has important theoretical values and wide application prospects. In decade years, many scholars have focused on optimal management problems for DC pension plans with different market assumptions and special situations, and achieved many important research results, for example, Boulier et al. (2001), Deelstra et al. (2003), Zhang & Ewald (2010), Korn et al. (2011), Han & Hung (2012), Guan & Liang (2014), Blake et al. (2014), Liang & Ma (2015), Konicz & Mulvey (2015) and Guan & Liang (2016). In these papers, Boulier et al. (2001) investigated a DC pension plan with a minimum guarantee in a Vasicek interest rate environment, where the guarantee depended on the benefits and the level of stochastic interest rate. Deelstra et al. (2003) studied the optimal investment strategy for a DC pension plan with a minimum guarantee in a stochastic affine interest rate environment including the Vasicek model and the CIR model as special cases, where the guarantee was supposed to be a random variable. Zhang & Ewald (2010) focused on optimal investment strategies for a DC pension fund in an inflation risk framework, where the contribution rate was a fixed share of his salary. Korn et al. (2011) considered the portfolio selection problem for a DC pension plan in a hidden Markov-modulated economy. Han & Hung (2012) concerned a DC pension management problem with a minimum guarantee in an inflation risk and interest rate risk framework, where the guarantee and the contribution rate depended on the values of interest rate and inflation. Guan & Liang (2014) assumed the minimum guarantee to be related with the random time of death and studied the optimal strategy for a DC plan in a stochastic interest rate and stochastic volatility framework. Blake et al. (2014) studied the optimal funding and investment strategy for a DC pension plan with desired profile consumption over the lifetime of members, where the planner has a rational life cycle and its contribution rate is age-dependent. Liang & Ma (2015) considered a DC pension management problem with mortality risk and salary risk. Konicz & Mulvey (2015) discussed optimal strategies for individuals with a DC pension plan and provided optimal decisions to manage pension savings before and after retirement. Guan & Liang (2016) focused on a Nash equilibrium strategy with inflation risk between two DC pension plans, where the contribution rate is closely related with inflation state. For more researches on DC pension plans, researches can read the works of Giacinto et al. (2011) and Zhang et al. (2013) and so forth. These papers provided some optimal strategies for pension fund managers to maximize their wealth, but these strategies were derived under the assumption of power utility, i.e. the risk preference of a fund manager was assumed to satisfy a power utility, which was a special utility function. Practically, the risk preference of a fund manager may satisfy other utilities, for example, the exponential utility or the logarithm utility. Therefore, the assumption of power utility is a drawback for these papers. In the utility theory, the hyperbolic absolute risk aversion (HARA) utility is a general utility function, which includes power utility, exponential utility and logarithm utility as special cases. It is obvious that optimal strategies under the HARA utility are more suitable for fund managers. However, if we extend the power utility in the above literature to a HARA utility in an incomplete market, there will be many difficulties in solving the optimal portfolios. So optimal portfolios with the HARA preference in an incomplete financial market have attracted more attentions in the portfolio selection problems and many scholars have been exploring new research methods. Nowadays, utility maximization problems with the HARA utility in the presence of complete financial markets have been solved in a number of papers and presented several methods, for example, Tepla (2001), Grasselli (2003), Vigna (2014) etc. Tepla (2001) presented a martingale method to study optimal portfolios with minimum performance constraints and optimal portfolios for a HARA-utility investor are obtained explicitly. Grasselli (2003) also used a martingale method to investigate a continuous-time portfolio selection problem with stochastic interest rate and HARA preference and verified the fact that optimal policies under HARA preference converged to almost surely those under the exponential preference or the logarithm preference. However, there are some difficulties in using the martingale method to solve some more complicated portfolio selection models, for example, the CEV model or the stochastic interest rate and stochastic volatility. In these difficulties, constructing an exponential martingale is one of most difficult factors. Vigna (2014) used the dynamic programming principle to study optimal portfolios with HARA preference as well and obtained explicit solutions by directly conjecturing the form of solution of the value function. Afterwards, Jonsson & Sircar (2002) first proposed Legendre transform-dual theory and used it to successfully solve some sophisticated portfolio selection models. Meanwhile, this work also laid down theoretical foundation for wide applications of Legendre transform-dual theory. For more results on Legendre transform-dual theory, readers may refer to the papers of Xiao et al. (2007), Gao (2008, 2010), Jung & Kim (2012), Chang & Chang (2014), Chang & Rong (2014) and Chang & Chang (2014) and so on. In these papers, Xiao et al. (2007) studied the optimal policy for a DC pension fund with logarithm preference under the CEV model. Gao (2008) investigated the optimal policy for a DC pension fund with stochastic affine interest rate under the logarithm preference. Gao (2010) solved a DC pension fund management problem with more general CEV model and presented Legendre transform-dual theory with four state variables. Jung & Kim (2012) considered an investment problem with HARA preference under the CEV model. Chang et al. (2014) focused on an asset liability management problem with affine interest rate under the HARA preference. Compared with the classical martingale method presented by Tepla (2001) and Grasselli (2003), Legendre transform-dual theory can be used in a more convenient and understandable form; more importantly, the difficulties in the applications of Legendre transform-dual theory are more easily overcome and more problems can be solved. Compared with those portfolio choice problems with HARA utility in a complete market, these in an incomplete market are more difficult to solve. This paper is concerned with a DC pension plan with a stochastic interest rate and stochastic returns under the HARA preference, which is actually an optimal investment problem in an incomplete market, and the principle of stochastic dynamic programming along with Legendre transform-dual theory is used to solve it successfully. In this paper, following the model of Guan & Liang (2015), we assume risk preference of a fund manager to satisfy a HARA utility and focus on a DC pension fund management problem with a stochastic interest rate and mean-reverting returns. The fund manager invests his wealth into the financial market with one risk-free asset, one stock and one rolling bond, where the interest rate is driven by a stochastic affine interest rate including the Vasicek (1977) and the Cox–Ingersoll–Ross (CIR; Cox et al., 1985) model as special cases. The return of the stock is supposed to be stochastic and satisfy a mean-reverting process, which was calibrated to U. S. stock market data in the paper of Viceira & Campbell (1999). It indicates that the stock price with mean-reverting returns is more suitable to characterize the financial market with features of bull and bear. In addition, the stock price dynamics are supposed to be simultaneously driven by random sources of interest rate and the stock market itself. The fund manager expects to seek an optimal policy to maximize the expected utility of terminal wealth. In order to derive optimal policies under the HARA preference, we introduce a Legendre transform-dual technique to solve this problem. After some deductions and calculations, we explicitly obtain closed-form solutions of optimal policies. Meanwhile, we also derive some special cases, including power utility, logarithm utility and exponential utility. A numerical simulation is provided to analyse the impact of market parameters on the optimal strategies. To sum up, this paper contains the following contributions: (i) a DC pension fund management problem with a stochastic interest rate and stochastic return is studied; (ii) optimal policies under the HARA utility in an incomplete financial market are explicitly obtained; (iii) the Legendre transform-dual technique with four state variables is introduced to deal with it; (iv) a numerical simulation is presented to illustrate our results. The reminder of this paper is organized as follows. Section 2 formulates the problem framework. Section 3 introduces Legendre transform-dual theory to transform the nonlinear Hamilton–Jocabi–Bellman (HJB) equation into its linear dual equation. Section 4 derives optimal portfolios for a fund manager with HARA preference and Section 5 gives some special cases to enrich and extend our model. Section 6 provides a numerical simulation to illustrate our results and Section 7 concludes this paper. 2. Problem formulation In this paper, there are some symbol contracts. We denote the complete probability space by |$(\varOmega ,\mathscr{F},\{\mathscr{F}_t \}_{t\in [0,T]},\mathbb{P})$|⁠, where |$\mathscr{F}_t $| stands for the information filtration available in the market, denote the mathematical expectation by |$\mathbb{E}(\cdot )$| and denote the fixed and finite horizon by |$[0,T]$|⁠. The price of riskless asset (i.e. a cash or bank account) is denoted by |$S_0 (t)$| at time |$t$|⁠, then |$S_0 (t)$| satisfies the following equation: $$\begin{equation} \frac{dS_0 (t)}{S_0 (t)}=r(t)dt, \quad S_0 (0)=1. \end{equation}$$ (1) In this paper, we suppose the interest rate |$r(t)$| to be stochastic, and |$r(t)$| evolves according to $$\begin{equation} dr(t)=(a-br(t))dt-\sqrt{k_1 r(t)+k_2} dW_r (t), \quad r(0)=r_0>0, \end{equation}$$ (2) where |$a,b,k_1 $| and |$k_2 $| are all positive constants and |$W_r (t)$| is a standard Brownian motion defined on |$(\varOmega ,\mathscr{F},\{\mathscr{F}_t \}_{t\in [0,T]},\mathbb{P})$|⁠. It needs the condition |$2a\ge k_1 $| to ensure |$r(t)\ge 0$| for any |$t\in [0,T]$|⁠. Remark 1 When |$k_1 =0$| or |$k_2 =0$|⁠, Equation (2) is degenerated into the Vasicek (1977) or the Cox et al. (1985) model. So we call Equation (2) as a stochastic affine interest rate model. This model has been studied by Deelstra et al. (2003), Gao (2008), Guan & Liang (2014, 2015) and Chang & Chang (2014). The risky asset is a stock, whose price is denoted by |$S_1(t)$| at time |$t$|⁠. Considering the fact that the stock price must be influenced by interest rate dynamics in the environment of stochastic interest rate, the return from stock dynamics denoted by |$L(t)$| is generally stochastic and mean-reverting. Therefore, the dynamics of |$S_1 (t)$| is supposed to satisfy the following stochastic differential equation (SDE) (cf. Guan & Liang, 2015) $$\begin{align}\frac{dS_1 (t)}{S_1 (t)}=&\ r(t)dt+\sigma _r \sqrt{k_1 r(t)+k_2} \left( {\lambda _r \sqrt{k_1 r(t)+k_2 } dt+dW_r (t)} \right)\nonumber\\ &+\sigma _s(L(t)dt+dW_s (t)), \quad S_1 (0)=s_1>0; \end{align}$$ (3) $$\begin{equation} \hskip-35ptdL(t)=\alpha (\delta -L(t))dt+\sigma _l dW_L (t), \quad L(0)=l_0>0; \end{equation}$$ (4) where, in Equation (3), |$\sigma _r,\lambda _r $| and |$\sigma _s $| are all positive constants, |$\sigma _r \sqrt{k_1 r(t)+k_2} $| and |$\sigma _s $| are volatilities caused by the interest rate dynamics and the stock dynamics itself and |$\lambda _r \sqrt{k_1 r(t)+k_2 } $| and |$L(t)$| are risk premiums caused by the interest rate risk and the stock risk, respectively. In Equation (4), |$\alpha ,\delta $| and |$\sigma _l $| are all positive constants, |$\alpha $| stands for the mean-reverting speed, |$\delta $| stands for the mean-reverting level and |$\sigma _l $| stands for the volatility caused by the dynamics of |$L(t)$|⁠. When |$\delta $| is higher, the market can be taken to be bull. Inversely, the market is bear when |$\delta $| is lower. Moreover, |$W_s (t)$| and |$W_L (t)$| are standard Brownian motions on |$(\varOmega ,\mathscr{F},\{\mathscr{F}_t \}_{t\in [0,T]},\mathbb{P})$| with |$Cov(W_s (t),W_L (t))=\rho _{sl}t $|⁠. Meantime, we assume that |$W_r (t)$| is independent of |$W_s (t)$| and |$W_L (t)$|⁠. From Equations (3) and (4), we can see the financial market is actually incomplete. Remark 2 The correlation coefficient |$\rho _{sl} $| has the following economic implications: (i) if |$\rho _{sl} =0$|⁠, it implies that |$W_L (t)$| is independent of |$W_s (t)$|⁠; (ii) if |$\rho _{sl} \in (-1,1)$|⁠, it means that the risk from stochastic return can’t be completely hedged by the stock; (iii) if |$\rho _{sl} =\pm 1$|⁠, |$W_L (t)$| is equivalent to |$W_s (t)$|⁠, it indicates that the risk from stochastic return can be completely hedged. In the financial market with stochastic interest rate, zero-coupon bonds are often issued to avoid the risk of interest rate and their prices are closely related with the dynamics of interest rate. We suppose that the price of a zero-coupon bond with maturity |$s$| at time |$t$| is denoted by |$B(t,s)$|⁠, then |$B(t,s)$| follows the following SDE (cf. Deelstra et al., 2003): $$\begin{equation} \frac{dB(t,s)}{B(t,s)}=r(t)dt+h_0 (s-t)\sqrt{k_1 r(t)+k_2} \left( {\lambda _r \sqrt{k_1 r(t)+k_2 } dt+dW_r (t)} \right), \end{equation}$$ (5) with the terminal condition |$B(s,s)=1$|⁠. And |$h_0 (t)$| is given by $$\begin{equation*} h_0 (t)=\frac{2(e^{mt}-1)}{m-(b-k_1 \lambda _r )+e^{mt}(m+b-k_1 \lambda _r )}, \quad m=\sqrt{(b-k_1 \lambda _r )^2+2k_1}. \end{equation*}$$ Further, the explicit solution to Equation (5) is given by $$\begin{equation} B(t,s)=\exp \big\{-h_0 (s-t)r(t)+h_1 (s-t)\big\}, \end{equation}$$ (6) where $$\begin{equation*} h_1 (t)=\frac{k_2} {k_1 }t-\frac{k_2 }{k_1 }h_0 (t)+\frac{2}{k_1 }(a+\frac{k_2 }{k_1 }b)\ln \frac{2me^{0.5(m+b-k_1 \lambda _r )t}}{m-(b-k_1 \lambda _r )+e^{mt}(m+b-k_1 \lambda _r )}. \end{equation*}$$ The maturity of a bond is given by |$s-t$|⁠, which continuously varies over time. Since it is possible for a fund manager not to find a zero-coupon bond with any maturity |$s>0$|⁠, it is unrealistic to invest in |$B(t,s)$|⁠. So it is necessary for us to introduce a rolling bond with a constant maturity |$K$|⁠, which we can invest to hedge the risk of interest rate. The price dynamics of the rolling bond |$B_K (t)$| can be supposed to satisfy the following SDE (cf. Boulier et al., 2001 and Guan & Liang, 2015): $$\begin{equation} \frac{dB_K (t)}{B_K (t)}=r(t)dt+h_0 (K)\sqrt{k_1 r(t)+k_2} \left( {\lambda _r \sqrt{k_1 r(t)+k_2 } dt+dW_r (t)} \right). \end{equation}$$ (7) As stated by Boulier et al. (2001) and Guan & Liang (2015), the rolling bond is only related with interest rate. Moreover, the zero-coupon bond can be replicated by the rolling bond and the cash in the market. The relation between the zero-coupon bond and the rolling bond is given by $$\begin{equation} \frac{dB(t,s)}{B(t,s)}=\left( {1-\frac{h_0 (s-t)}{h_0 (K)}} \right)\frac{dS_0 (t)}{S_0 (t)}+\frac{h_0 (s-t)}{h_0 (K)}\cdot \frac{dB_K (t)}{B_K (t)}. \end{equation}$$ (8) So the rolling bond is a more generic asset. Consider a pension participant who enters a pension plan at time |$0$| with initial wealth |$x_0>0$|⁠. He continuously contributes for |$T$| years and retires at time |$T$|⁠. In order to preserve and increase the value of a pension fund, he invests his pension fund into the above described financial market. Suppose that the contribution rate in a DC pension plan is a constant and is denoted by |$C$|⁠. Let the proportions of money invested in the stock and rolling bond be denoted by |$\pi _s (t)$| and |$\pi _B (t)$|⁠, respectively, then the proportion invested in the cash is denoted by |$\pi _0 (t)=1-\pi _s (t)-\pi _B (t)$|⁠. There is no transaction cost or taxes in the financial market and short selling or buying is allowed. Under the strategies |$\pi _s (t)$| and |$\pi _B (t)$|⁠, the wealth of pension plan is given by $$\begin{align}dX(t)=&\ \left( {r(t)X(t)+\pi _s (t)\sigma _s X(t)L(t)+C} \right)dt \nonumber\\&+\lambda _r \left( {\pi _B (t)h_0 (K)+\pi _s (t)\sigma _r} \right)(k_1 r(t)+k_2 )X(t)dt+\pi _s (t)\sigma _s X(t)dW_s (t)\nonumber \\&+X(t)\left( {\pi _B (t)h_0 (K)+\pi _s (t)\sigma _r } \right)\sqrt{k_1 r(t)+k_2 } dW_r (t),\end{align}$$ (9) with the initial condition given by |$X(0)=x_0>0$|⁠. Definition 1 The strategies |$\pi _s (t)$| and |$\pi _B (t)$| are said to be admissible, if the following three conditions are met: (i) |$\pi _s (t)$| and |$\pi _B (t)$| are |$\mathscr{F}_t $|- progressively measurable; (ii) |$\mathbb{E}(\int _0^T {X^2(t)} \left ( {\pi _B (t)h_0 (K)+\pi _s (t)\sigma _r} \right )^2(k_1 r(t)+k_2 )\mathrm{d}t+\int _0^T {\pi _s^2 (t)\sigma _s^2 X^2(t)\mathrm{d}t} )<\infty $|⁠; (iii) Equation (9) has the unique solution for any |$\pi _s (t)$| and |$\pi _B (t)$|⁠. Denoting the set of all admissible strategies by |$\varGamma =\left \{ {\pi (t)=(\pi _s (t),\pi _B (t))\left | {0\le t\le T} \right .} \right \}$|⁠, the fund manager hopes to obtain the optimal strategy |$\pi (t)\in \varGamma $| to maximize the expected utility of terminal wealth in the above pension plan, so the objective function can be mathematically formulated as follows: $$\begin{equation} \mathop{\sup} \limits_{\pi \left( t \right)\in \varGamma } \mathbb{E}\left[{U\left( {X\left( T \right)} \right)} \right], \end{equation}$$ (10) where |$U(\cdot )$| is a utility function, which can describe the satisfactory degree of a fund manager for risks. In this paper, we discuss the solution of the problem (10) in a general utility framework, where the utility function is called a HARA utility function and it is given by $$\begin{equation} U(x)=U(\eta,p,q,x)=\frac{1-p}{qp}\left( {\frac{q}{1-p}x+\beta} \right)^p, \quad q>0, \quad p<1, \quad p\ne 0. \end{equation}$$ (11) 3. HJB equation and Legendre transform This section uses a Legendre transform-dual theory to transform the HJB equation from the principle of stochastic dynamic programming into its dual one. We define the value function |$H\left ( {t,r,l,x} \right )$| as follows: $$\begin{equation*} H\left( {t,r,l,x} \right)=\mathop{\sup} \limits_{\pi \left( t \right)\in \varGamma } \mathbb{E}\big[{U\left( {X\left( T \right)} \right)\vert r\left( t \right)=r,L\left( t \right)=l,X\left( t \right)=x} \big], \end{equation*}$$ with the boundary condition given by |$H\left ( {T,r,l,x} \right )=U\left ( x \right )$|⁠. According to the principle of stochastic dynamic programming, |$H(t,r,l,x)$| satisfies the following HJB equation: $$\begin{align} \mathop{\sup} \limits_{\pi \left( t \right)\in \varGamma } &\Big\{ {H_t +\left( {rx+x\lambda _r \left( {\pi _B \left( t \right)h_0 \left( K \right)+\pi _s \left( t \right)\sigma _r } \right)\left( {k_1 r+k_2 } \right)+\pi _s \left( t \right)\sigma _s xl+C} \right)} H_x \nonumber\\& \ +\frac{1}{2}x^2\left( {\pi _B \left( t \right)h_0 \left( K \right)+\pi _s \left( t \right)\sigma _r } \right)^2\left( {k_1 r+k_2 } \right)H_{xx} +\frac{1}{2}\left( {\pi _s \left( t \right)\sigma _s x} \right)^2H_{xx}\nonumber \\&\ +\left( {a-br} \right)H_r +\frac{1}{2}\left( {k_1 r+k_2 } \right)H_{rr} +\alpha \left( {\delta -l} \right)H_l +\frac{1}{2}\sigma _l^2 H_{ll} \nonumber\\& \ -\left( {\pi _B \left( t \right)h_0 \left( K \right)+\pi _s \left( t \right)\sigma _r } \right)\left( {k_1 r+k_2 } \right)xH_{xr} {+\rho _{sl} \pi _s \left( t \right)\sigma _s \sigma _l xH_{xl}} \Big\} =0,\end{align}$$ (12) where |$H_t $|⁠, |$H_r $|⁠, |$H_{rr} $|⁠, |$H_l $|⁠, |$H_{ll} $|⁠, |$H_x $|⁠, |$H_{xx} $|⁠, |$H_{xr} $| and |$H_{xl} $| denote the first-order and second-order partial differential derivatives with respect to the variables |$t$|⁠, |$r$|⁠, |$l$| and |$x$|⁠. Differentiating Equation (12) with respect to |$\pi _s (t)$| and |$\pi _B (t)$|⁠, we obtain $$\begin{align} \hskip-115pt\pi _s^\ast \left( t \right)=-\frac{l}{\sigma _s x}\cdot \frac{H_x} {H_{xx} }-\frac{\rho _{sl} \sigma _l }{\sigma _s x}\cdot \frac{H_{xl}}{H_{xx}}, \end{align}$$ (13) $$\begin{align} \pi _B^\ast \left( t \right)=\frac{l\sigma _r -\lambda _r \sigma _s }{\sigma _s h_0 \left( K \right)x}\cdot \frac{H_x }{H_{xx}} +\frac{1}{h_0 \left( K\right)x}\cdot \frac{H_{xr}}{H_{xx}}+\frac{\rho _{sl} \sigma _l \sigma _r }{\sigma _s h_0 \left( K \right)x}\cdot \frac{H_{xl}}{H_{xx}}. \end{align}$$ (14) Substituting Equations (13) and (14) back into Equation (12), we get $$\begin{align} &H_t +\left( {rx+C} \right)H_x +\left( {a-br} \right)H_r +\frac{1}{2}\left( {k_1 r+k_2} \right)H_{rr} +\alpha \left( {\delta -l} \right)H_l\nonumber \\&\quad +\frac{1}{2}\sigma _l^2 H_{ll} -\frac{1}{2}\left( {\lambda _r^2 \left( {k_1 r+k_2 } \right)+l^2} \right)\frac{H_x^2 }{H_{xx}}+\lambda _r \left( {k_1 r+k_2 } \right)\frac{H_x H_{xr}}{H_{xx}}\nonumber \\&\quad -\frac{1}{2}\left( {k_1 r+k_2 } \right)\frac{H_{xr}^2 }{H_{xx} }-\frac{1}{2}\rho _{sl}^2 \sigma _l^2 \frac{H_{xl}^2 }{H_{xx}}-\rho _{sl} \sigma _l l\frac{H_x H_{xl}}{H_{xx}}=0,\end{align}$$ (15) with the boundary condition given by |$H\left ( {T,r,l,x} \right )=\frac{1-p}{qp}\big ( {\frac{q}{1-p}x+\beta } \big)^p$|⁠. Considering the fact that the structure of the solution to Equation (15) is difficult to directly conjecture, we introduce the following Legendre transform and convert Equation (15) into its dual equation. Definition 2 Suppose |$f:{\mathbb{R}}^n\to{\mathbb{R}}$| to be a concave function, Legendre transform can be defined as $$\begin{equation*} L\left( z \right)=\mathop{\max }\limits_x \left\{ \,{f\left( x \right)-zx} \right\}, \end{equation*}$$ then |$L\left ( z \right )$| is called the dual function to |$f\left ( x \right )$|(cf. Jonsson & Sircar, 2002 and Gao, 2008). We define Legendre transform on |$H\left ( {t,r,l,x} \right )$| as follows: $$\begin{align} \hskip-70pt\hat{H}\left( {t,r,l,z} \right)=\mathop{\sup }\limits_{x>0} \left\{ {H\left( {t,r,l,x} \right)-zx} \right\}, \end{align}$$ (16) $$\begin{align} g\left( {t,r,l,z} \right)=\mathop{\inf }\limits_{x>0} \left\{ {x\vert H\left( {t,r,l,x} \right)\ge zx+\hat{H}\left( {t,r,l,z} \right)} \right\}, \end{align}$$ (17) where |$z>0$| is the dual variable to |$x$|⁠. So the relationship between |$g\left ( {t,r,l,z} \right )$| and |$\hat{H}\left ( {t,r,l,z} \right )$| is given by (cf. Jonsson & Sircar, 2002) $$\begin{equation} g\left( {t,r,l,z} \right)=-\hat{H}_z \left( {t,r,l,z} \right). \end{equation}$$ (18) Here both |$g\left ( {t,r,l,z} \right )$| and |$\hat{H}\left ( {t,r,l,z} \right )$| are dual functions to |$H\left ( {t,r,l,x} \right )$|⁠. For convenience, we choose |$g\left ( {t,r,l,z} \right )$| for our analysis, and we have $$\begin{equation} H\left( {t,r,l,z} \right)=H\left( {t,r,l,g} \right)-zg, \quad g\left( {t,r,l,z} \right)=x. \end{equation}$$ (19) Inspired by the works of Jonsson & Sircar (2002) and Gao (2010), after some deductions, we get the following rules: $$\begin{align} H_x =&\ z, \quad H_t =\hat{H}_t, \quad H_r =\hat{H}_r, \quad H_l =\hat{H}_l, \quad H_{xx} =-\frac{1}{\hat{H}_{zz}},\nonumber\\ H_{rr} =&\ \hat{H}_{rr} -\frac{\hat{H}_{rz}^2} {\hat{H}_{zz}}, \quad H_{ll} =\hat{H}_{ll} -\frac{\hat{H}_{lz}^2 }{\hat{H}_{zz}}, \quad H_{xr} =-\frac{\hat{H}_{rz}}{\hat{H}_{zz}}, \quad H_{xl} =-\frac{\hat{H}_{lz}}{\hat{H}_{zz}}. \end{align}$$ (20) According to |$H\left ( {T,r,l,x} \right )=U\left ( x \right )$|⁠, at the terminal time |$T$|⁠, we have $$\begin{equation} \begin{aligned} \hat{H}\left( {T,r,l,z} \right)=&\ \mathop{\sup} \limits_{x>0} \left\{ {U\left( x \right)-zx} \right\}, \\ g\left( {T,r,l,z} \right)=&\ \mathop{\inf }\limits_{x>0} \left\{ {x\vert U\left( x \right)\ge zx+\hat{H}\left( {T,r,l,z} \right)} \right\}. \end{aligned} \end{equation}$$ (21) So we get |$g\left ( {T,r,l,z} \right )=\left ( {{U}^{\prime}} \right )^{-1}\left ( z \right )$|⁠, where |$\left ( {{U}^{\prime}} \right )^{-1}\left ( z \right )$| is the inverse of marginal utility. Plugging Equation (20) into Equation (15), we get $$\begin{equation} \begin{aligned} \hat{H}_t &+rgz+Cz+\left( {a-br} \right)\hat{H}_r +\frac{1}{2}\left( {k_1 r+k_2} \right)\hat{H}_{rr} +\alpha \left( {\delta -l} \right)\hat{H}_l \\& +\frac{1}{2}\sigma _l^2 \hat{H}_{ll} -\rho _{sl} \sigma _l lz\hat{H}_{lz} +\frac{1}{2}\left( {\lambda _r^2 \left( {k_1 r+k_2 } \right)+l^2} \right)z^2\hat{H}_{zz} \\& +\lambda _r \left( {k_1 r+k_2 } \right)z\hat{H}_{rz} +\frac{1}{2}\sigma _l^2 \left( {\rho _{sl}^2 -1} \right)\frac{\hat{H}_{lz}^2 }{\hat{H}_{zz}}=0. \end{aligned} \end{equation}$$ (22) Differentiating Equation (22) with respect to |$z$| and using Equation (18), we obtain $$\begin{equation} \begin{aligned} g_t &-rg-C+\left( {\lambda _r^2 \left( {k_1 r+k_2} \right)+l^2-r} \right)zg_z +\left( {a-br+\lambda _r \left( {k_1 r+k_2 } \right)} \right)g_r \\& +\frac{1}{2}\left( {k_1 r+k_2 } \right)g_{rr} +\left( {\alpha \left( {\delta -l} \right)-\rho _{sl} \sigma _l l} \right)g_l +\frac{1}{2}\left( {\lambda _r^2 \left( {k_1 r+k_2 } \right)+l^2} \right)z^2g_{zz} \\& +\frac{1}{2}\sigma _l^2 g_{ll} +\lambda _r \left( {k_1 r+k_2 } \right)zg_{rz} -\rho _{sl} \sigma _l lzg_{lz} +\frac{1}{2}\sigma _l^2 \left( {\rho _{sl}^2 -1} \right)\left( {\frac{2g_l g_{lz}}{g_z }-\frac{g_l^2 g_{zz}}{g_z^2 }} \right)=0, \end{aligned} \end{equation}$$ (23) with the boundary condition given by |$g\left ( {T,r,l,z} \right )=\frac{1-p}{q}\big( {z^{\frac{1}{p-1}}-\beta } \big)$|⁠. 4. The optimal portfolios This section uses a variable change technique to solve Equation (23), and derive optimal policies for the problem (10). The solution to Equation (23) can be conjectured as $$\begin{equation} g\left( {t,r,l,z} \right)=\frac{1-p}{q}z^{\frac{1}{p-1}}f\left( {t,r,l} \right)-\frac{1-p}{q}\beta h\left( {t,r} \right)+J\left( {t,r} \right), \end{equation}$$ (24) with boundary conditions |$f\left ( {T,r,l} \right )=1$|⁠, |$h\left ( {T,r} \right )=1$| and |$J\left ( {T,r} \right )=0$|⁠. The partial derivatives with respect to |$g\left ( {t,r,l,z} \right )$| are as follows: $$\begin{align} g_t =&\ \frac{1-p}{q}z^{\frac{1}{p-1}}f_t -\frac{1-p}{q}\beta h_t +J_t, \quad g_{zz} =-\frac{1}{q}\cdot \frac{2-p}{p-1}z^{\frac{3-2p}{p-1}}f,\nonumber\\ g_z =&\ -\frac{1}{q}z^{\frac{2-p}{p-1}}f, \quad g_r =\frac{1-p}{q}z^{\frac{1}{p-1}}f_r -\frac{1-p}{q}\beta h_r +J_r,\nonumber\\ g_{rr} =&\ \frac{1-p}{q}z^{\frac{1}{p-1}}f_{rr} -\frac{1-p}{q}\beta h_{rr} +J_{rr}, \quad g_{ll} =\frac{1-p}{q}z^{\frac{1}{p-1}}f_{ll},\nonumber\\ g_l =&\ \frac{1-p}{q}z^{\frac{1}{p-1}}f_l, \quad g_{rz} =-\frac{1}{q}z^{\frac{2-p}{p-1}}f_r, \quad g_{lz} =-\frac{1}{q}z^{\frac{2-p}{p-1}}f_l. \end{align}$$ (25) Putting Equations (24) and (25) into Equation (23), after some simplifications, we get $$\begin{equation*} \begin{split} &\frac{1-p}{q}z^{\frac{1}{p-1}}\left[{f_t +\left( {\frac{p}{2\left( {1-p} \right)^2}\lambda _r^2 \left( {k_1 r+k_2} \right)+\frac{p}{2\left( {1-p} \right)^2}l^2+\frac{p}{1-p}r} \right)} \right.f \\& \quad\quad\quad\quad\quad+\left( {a-br-\frac{p}{1-p}\lambda _r \left( {k_1 r+k_2 } \right)} \right)f_r +\left( {\alpha \left( {\delta -l} \right)+\frac{p}{1-p}\rho _{sl} \sigma _l l} \right)f_l \\& \quad\quad\quad\quad\quad+\frac{1}{2}\sigma _l^2 f_{ll} +\frac{1}{2}\left( {k_1 r+k_2 } \right)f_{rr} \left. {+\frac{1}{2}\sigma _l^2 \left( {\rho _{sl}^2 -1} \right)p\frac{f_l^2 }{f}} \right] \\& \quad-\frac{1-p}{q}\beta \left[{h_t -rh+\left( {a-br+\lambda _r \left( {k_1 r+k_2 } \right)} \right)h_r +\frac{1}{2}\left( {k_1 r+k_2 } \right)h_{rr}} \right] \\& \quad\quad\quad\quad\quad\quad+J_t -rJ+\left( {a-br+\lambda _r \left( {k_1 r+k_2 } \right)} \right)J_r +\frac{1}{2}\left( {k_1 r+k_2 } \right)J_{rr} -C=0. \end{split} \end{equation*}$$ Eliminating the dependence on |$z$| and |$\beta $|⁠, we obtain the following three PDEs: $$\begin{equation} \begin{aligned} f_t &+\left( {\frac{p}{2\left( {1-p} \right)^2}\lambda _r^2 \left( {k_1 r+k_2} \right)+\frac{p}{2\left( {1-p} \right)^2}l^2+\frac{p}{1-p}r} \right)f \\& +\left( {a-br-\frac{p}{1-p}\lambda _r \left( {k_1 r+k_2 } \right)} \right)f_r +\left( {\alpha \left( {\delta -l} \right)+\frac{p}{1-p}\rho _{sl} \sigma _l l} \right)f_l \\& +\frac{1}{2}\sigma _l^2 f_{ll} +\frac{1}{2}\left( {k_1 r+k_2 } \right)f_{rr} +\frac{1}{2}\sigma _l^2 \left( {\rho _{sl}^2 -1} \right)p\frac{f_l^2 }{f}=0,\;\;f\left( {T,r,l} \right)=1; \end{aligned} \end{equation}$$ (26) $$\begin{equation} J_t -rJ+\left( {a-br+\lambda _r \left( {k_1 r+k_2} \right)} \right)J_r +\frac{1}{2}\left( {k_1 r+k_2 } \right)J_{rr} -C=0, \quad J\left( {T,r} \right)=0; \end{equation}$$ (27) $$\begin{equation} h_t -rh+\left( {a-br+\lambda _r \left( {k_1 r+k_2} \right)} \right)h_r +\frac{1}{2}\left( {k_1 r+k_2 } \right)h_{rr} =0, \quad h\left( {T,r} \right)=1. \end{equation}$$ (28) In order to read conveniently, we describe the solving processes of Equations (26)–(28) as following lemmas. Lemma 1 Assume the solution to Equation (26) to be |$f\left ( {t,r,l} \right )=e^{A_1 \left ( t \right )+A_2 \left ( t \right )r+A_3 \left ( t \right )l+A_4 \left ( t \right )l^2}$|⁠, with boundary conditions given by |$A_1 \left ( T \right )=0$|⁠, |$A_2 \left ( T \right )=0$|⁠, |$A_3 \left ( T \right )=0$| and |$A_4 \left ( T \right )=0$|⁠, then |$A_3 (t)$| and |$A_1 (t)$| are respectively given by $$\begin{equation} \hskip-220ptA_3 \left( t \right)=\left( {-2\alpha \delta} \right)\int_t^T{A_4 \left( s \right)} e^{\int_t^s{\xi (z)dz}}\mathrm{d}s, \end{equation}$$ (29) $$\begin{equation} \begin{aligned} A_1 \left( t \right)=&\ \left( {a-\frac{p}{1-p}\lambda _r k_2} \right)\int_t^T{A_2 \left( s \right)\mathrm{d}s} +\frac{1}{2}k_2 \int_t^T{A_2^2 \left( s \right)\mathrm{d}s} +\alpha \delta \int_t^T{A_3 \left( s \right)\mathrm{d}s} \\& +\frac{1}{2}\sigma _l^2 \left( {1+\left( {\rho _{sl}^2 -1} \right)p} \right)\int_t^T{A_3^2 \left( s \right)\mathrm{d}s} \mbox{+}\sigma _l^2 \int_t^T{A_4 \left( s \right)\mathrm{d}s} \mbox{+}\frac{p}{2\left( {1-p} \right)^2}\lambda _r^2 k_2 (T-t); \end{aligned} \end{equation}$$ (30) further, we discuss |$A_2 (t)$| and |$A_4 (t)$| in the following two cases: (i) if |$p<\min \left \{ {\frac{b^2}{\left ( {\lambda _r k_1 -b} \right )^2+2k_1 },\frac{\alpha ^2}{\sigma _l^2 +2\rho _{sl} \sigma _l \alpha +\alpha ^2}, 1} \right \}$| and |$p\ne 0$|⁠, i.e. |$\varDelta _1>0$| and |$\varDelta _2>0$|⁠, we get $$\begin{equation} A_2 \left( t \right)=\frac{m_1 m_2 \left( {1-e^{\sqrt{\varDelta _1} \left( {T-t} \right)}} \right)}{m_1 -m_2 e^{\sqrt{\varDelta _1 } \left( {T-t} \right)}}, \end{equation}$$ (31) $$\begin{equation} A_4 \left( t \right)=\frac{m_4 m_5 \left( {1-e^{\sqrt{\varDelta _2} \left( {T-t} \right)}} \right)}{m_4 -m_5 e^{\sqrt{\varDelta _2 } \left( {T-t} \right)}}; \end{equation}$$ (32) (ii) if |$\max \left \{ {\frac{b^2}{\left ( {\lambda _r k_1 -b} \right )^2+2k_1 },\frac{\alpha ^2}{\sigma _l^2 +2\rho _{sl} \sigma _l \alpha +\alpha ^2}} \right \}<p<1$|⁠, i.e. |$\varDelta _1<0$| and |$\varDelta _2<0$|⁠, we obtain $$\begin{equation} A_2 \left( t \right)=m_3 -\frac{\sqrt{-\varDelta _1} }{k_1 }\tan \left( {\arctan \frac{k_1 m_3 }{\sqrt{-\varDelta _1 }}-\frac{\sqrt{-\varDelta _1}}{2 }\left( {T-t} \right)} \right), \end{equation}$$ (33) $$\begin{equation} A_4 \left( t \right)=m_6 \mbox{+}\frac{\sqrt{-\varDelta _2} }{2q_2 }\tan \left( {\arctan \frac{-2q_2 m_6 }{\sqrt{-\varDelta _2 }}-\frac{\sqrt{-\varDelta _2} }{2}\left( {T-t} \right)} \right); \end{equation}$$ (34) where $$\begin{equation} \begin{split} \xi (t)=&\ \frac{p}{1-p}\rho _{sl} \sigma _l -\alpha +2\sigma _l^2 \left( {1+\left( {\rho _{sl}^2 -1} \right)p} \right)A_4 \left( t \right), \\ q_{2}=&\ -2\sigma _l^2 \left( {1+\left( {\rho _{sl}^2 -1} \right)p}\right), \end{split} \end{equation}$$ (35) $$\begin{equation} m_{1,2} =\frac{-(b+\frac{p}{1-p}\lambda _r k_1) \pm \sqrt{\varDelta _1}}{-k_1}, \quad m_3 =\frac{b+\frac{p}{1-p}\lambda _r k_1 }{k_1 }, \end{equation}$$ (36) $$\begin{equation}\hskip-93pt m_{4,5} =\frac{2(\frac{p}{1-p}\rho _{sl} \sigma _l -\alpha) \pm\sqrt{\varDelta _2}}{-4\sigma _l^2 \left( {1+\left( {\rho _{sl}^2 -1} \right)p} \right)}, \end{equation}$$ (37) $$\begin{equation} \hskip-55ptm_6 =\frac{\frac{p}{1-p}\rho _{sl} \sigma _l -\alpha} {-2\sigma _l^2 \left( {1+\left( {\rho _{sl}^2 -1} \right)p} \right)}, \end{equation}$$ (38) $$\begin{equation} \begin{aligned} \varDelta _1 =&\ \frac{1}{1-p}\left( {b^2-p\left( {\left( {\lambda _r k_1 -b} \right)^2+2k_1} \right)} \right), \\ \varDelta _2 =&\ \frac{1}{1-p}\left( {\alpha ^2-p\left( {\sigma _l^2 +2\rho _{sl} \sigma _l \alpha +\alpha ^2} \right)} \right). \end{aligned} \end{equation}$$ (39) Proof. Plugging |$f\left ( {t,r,l} \right )=e^{A_1 \left ( t \right )+A_2 \left ( t \right )r+A_3 \left ( t \right )l+A_4 \left ( t \right )l^2}$| into Equation (26), after some simplifications and separating variables, we get the following four ordinary differential equations (ODEs) $$\begin{equation} {A}^{\prime}_2 \left( t \right)+\frac{p}{2\left( {1-p} \right)^2}\lambda _r^2 k_1 +\frac{p}{1-p}-\left( {b+\frac{p}{1-p}\lambda _r k_1} \right)A_2 \left( t \right)+\frac{1}{2}k_1 A_2^2 \left( t \right)\mbox{=}0, \quad A_2 \left( T \right)=0; \end{equation}$$ (40) $$\begin{equation} {A}^{\prime}_4 \left( t \right)+\frac{p}{2\left( {1-p} \right)^2}+2\left( {\frac{p}{1-p}\rho _{sl} \sigma _l -\alpha} \right)A_4 \left( t \right)+2\sigma _l^2 \left( {1+\left( {\rho _{sl}^2 -1} \right)p} \right)A_4^2 \left( t \right)\mbox{=}0, \quad A_4 \left( T \right)=0; \end{equation}$$ (41) $$\begin{equation} {A}^{\prime}_3 \left( t \right)+2\alpha \delta A_4 \left( t \right)+\left( {\frac{p}{1-p}\rho _{sl} \sigma _l -\alpha} \right)A_3 \left( t \right)+2\sigma _l^2 \left( {1+\left( {\rho _{sl}^2 -1} \right)p} \right)A_3 \left( t \right)A_4 \left( t \right)\mbox{=}0, \quad A_3 \left( T \right)=0; \end{equation}$$ (42) $$\begin{equation} \begin{aligned} {A}^{\prime}_1 \left( t \right)&+\frac{p}{2\left( {1-p} \right)^2}\lambda _r^2 k_2 +\left( {a-\frac{p}{1-p}\lambda _r k_2} \right)A_2 \left( t \right)+\frac{1}{2}k_2 A_2^2 \left( t \right)+\sigma _l^2 A_4 \left( t \right) \\& +\alpha \delta A_3 \left( t \right)+\frac{1}{2}\sigma _l^2 \left( {1+\left( {\rho _{sl}^2 -1} \right)p} \right)A_3^2 \left( t \right)=0,\;\;A_1 \left( T \right)=0. \end{aligned} \end{equation}$$ (43) Applying the classic methods on ODEs, solving Equations (41)–(44), we get Equations (29)–(34). Lemma 2 Suppose that the solution to Equation (27) is given by |$J\left ( {t,r} \right )=-C\int _t^T {\hat{J}\left ( {s,r} \right )\mathrm{d}s} $|⁠, then |$\hat{J}\left ( {t,r} \right )$| meets the following PDE: $$\begin{equation} \hat{J}_t -r\hat{J}+\left( {a-br+\lambda _r \left( {k_1 r+k_2} \right)} \right)\hat{J}_r +\frac{1}{2}\left( {k_1 r+k_2 } \right)\hat{J}_{rr} =0, \quad \hat{J}\left( {T,r} \right)=1. \end{equation}$$ (44) Proof. Introducing the following variational operator for any function |$J\left ( {t,r} \right )$|⁠: $$\begin{equation*} \nabla J\left( {t,r} \right)=-rJ+\left( {a-br+\lambda _r \left( {k_1 r+k_2} \right)} \right)J_r +\frac{1}{2}\left( {k_1 r+k_2 } \right)J_{rr}, \end{equation*}$$ we rewrite Equation (27) as $$\begin{equation} \frac{\partial J\left( {t,r} \right)}{\partial t}+\nabla J\left( {t,r} \right)-C=0, \quad J\left( {T,r} \right)=0. \end{equation}$$ (45) Moreover, we obtain $$\begin{equation} \frac{\partial J\left( {t,r} \right)}{\partial t}=-C\left( {\int_t^T{\frac{\partial \hat{J}\left( {s,r} \right)}{\partial s}\mathrm{d}s-\hat{J}\left( {t,r} \right)}} \right), \end{equation}$$ (46) $$\begin{equation} \hskip-50pt\nabla J\left( {t,r} \right)=-C\int_t^T{\nabla \hat{J}\left( {s,r} \right)} \mathrm{d}s. \end{equation}$$ (47) Putting Equations (46) and (47) in Equation (45), we get $$\begin{equation} -C\left( {\int_t^T{\left( {\frac{\partial \hat{J}\left( {s,r} \right)}{\partial s}+\nabla \hat{J}\left( {s,r} \right)} \right)\mathrm{d}s-\hat{J}\left( {T,r} \right)+1}} \right)=0. \end{equation}$$ (48) So we have $$\begin{equation*} \frac{\partial \hat{J}\left( {s,r} \right)}{\partial s}+\nabla \hat{J}\left( {s,r} \right)=0, \quad \hat{J}\left( {T,r} \right)=1. \end{equation*}$$ Namely, Equation (44) holds. Lemma 3 Assume the solution to Equation (44) to satisfy the structure |$\hat{J}\left ( {t,r} \right )=e^{A_5 \left ( t \right )+A_6 \left ( t \right )r}$|⁠, with its boundary conditions |$A_5 \left ( T \right )=0$| and |$A_6 \left ( T \right )=0$|⁠, then |$A_6 \left ( t \right )$| and |$A_5 \left ( t \right )$| are respectively given by $$\begin{equation} \hskip-230ptA_6 \left( t \right)=\frac{m_7 m_8 \left( {1-e^{\sqrt{\varDelta _3} \left( {T-t} \right)}} \right)}{m_7 -m_8 e^{\sqrt{\varDelta _3 } \left( {T-t} \right)}}, \end{equation}$$ (49) $$\begin{align} &A_5 \left( t \right)=\frac{ak_1 +bk_2 }{k_1 }\left( {m_8 (T-t)+\frac{2}{k_1 }\ln \frac{m_7 -m_8 }{m_7 -m_8 e^{\sqrt{\varDelta _3 } \left( {T-t} \right)}}} \right)+\frac{k_2 }{k_1 }A_6 \left( t \right)+\frac{k_2 }{k_1 }(T-t), \end{align}$$ (50) where $$\begin{equation*} m_{7,8} =\frac{\lambda _r k_1-b }{-k_1 }\pm \frac{\sqrt{\varDelta _3 }}{-k_1 }, \quad \varDelta _3 =\left( {\lambda _r k_1 -b} \right)^2+2k_1>0. \end{equation*}$$ Proof. Substituting |$\hat{J}\left ( {t,r} \right )=e^{A_5 \left ( t \right )+A_6 \left ( t \right )r}$| back into Equation (44), after eliminating the dependence on |$r$|⁠, we get $$\begin{equation} {A}^{\prime}_6 \left( t \right)-1+\left( {\lambda _r k_1 -b} \right)A_6 \left( t \right)+\frac{1}{2}k_1 A_6^2 \left( t \right)\mbox{=}\ 0, \quad A_6 \left( T \right)=0. \end{equation}$$ (51) $$\begin{equation} {A}^{\prime}_5 \left( t \right)+\left( {a+\lambda _r k_2} \right)A_6 \left( t \right)+\frac{1}{2}k_2 A_6^2 \left( t \right)\mbox{=}\ 0, \quad A_5 \left( T \right)=0. \end{equation}$$ (52) Solving Equation (51), we can get Equation (49). By using (52)|$\times k_1 -$|(51)|$\times k_2 $|⁠, we can obtain $$\begin{equation*} k_{_1} {A}^{\prime}_5 \left( t \right)-k_2 {A}^{\prime}_6 \left( t \right)+k_2 +\left( {ak_{_1 } +bk_2 } \right)A_6 \left( t \right)\mbox{=}0. \end{equation*}$$ Solving the equation, we can obtain (50). Lemma 4 Conjecture that the solution to Equation (28) is given by |$h\left ( {t,r} \right )=e^{A_{7} \left ( t \right )+A_{8} \left ( t \right )r}$|⁠, with its boundary conditions given by |$A_7 \left ( T \right )=0$| and |$A_{8} \left ( T \right )=0$|⁠, then |$A_{7} \left ( t \right )=A_{5} \left ( t \right )$| and |$A_{8} \left ( t \right )=A_{6} \left ( t \right )$|⁠. Proof. We verify that the solution to Equation (28) is same as that to Equation (45). According to (20), (24) and (25), we get $$\begin{align*} \frac{H_{x}}{H_{xx}}=&\ -z\hat{H}_{zz} =zg_{z} =-\frac{1}{q}z^{\frac{1}{p-1}}f=-\frac{1}{1-p}\left( {x+\frac{1-p}{q}\beta h-J} \right),\\ \frac{H_{xr}}{H_{xx}}=&\ \hat{H}_{rz} =-g_r =-A_2 \left( t \right)x+\frac{1-p}{q}\beta h\left( A_{8} \left( t \right)-A_{2} \left( t \right) \right)+A_{2} \left( t \right)J-J_{r},\\ \frac{H_{xl}}{H_{xx}}=\hat{H}_{lz} =&\ -g_l =-\left( A_{3} \left( t \right)+2A_{4} \left( t \right)l \right)\left( {x+\frac{1-p}{q}\beta h-J} \right). \end{align*}$$ Finally, we summarize the above deductions as follows. Theorem 1 Assume that the utility function is given by the HARA utility (11), then the optimal strategy of the problem (10) is given by $$\begin{equation} \pi _s^\ast \left( t \right)=\frac{1}{X(t)\sigma _s} \left( {\frac{L(t)}{1-p}+\rho _{sl} \sigma _l \left( {A_3 \left( t \right)+2A_4 \left( t \right)L(t)} \right)} \right)\left( {X(t)+\frac{1-p}{q}\beta h(t,r)-J(t,r)} \right), \end{equation}$$ (53) $$\begin{equation} \begin{aligned} \hskip-28pt\pi _B^\ast \left( t \right)=&\ -\frac{1}{\sigma _s X(t)h_0 \left( K \right)}\left( {\frac{\sigma _r L(t)-\lambda _r \sigma _s} {1-p}+\rho _{sl} \sigma _l \sigma _r \left( {A_3 \left( t \right)+2A_4 \left( t \right)L(t)} \right)+\sigma _s A_2 \left( t \right)} \right) \\& \times \left( {X(t)+\frac{1-p}{q}\beta h(t,r)-J(t,r)} \right)+\frac{1}{X(t)h_0 \left( K \right)}\left( {\frac{1-p}{q}\beta A_8 \left( t \right)h(t,r)-J_r } \right), \end{aligned} \end{equation}$$ (54) with the optimal value function given by $$\begin{equation} H_{HARA}^\ast (t,r,l,x)=\frac{1-p}{qp}\left( {\frac{q}{1-p}(x-J(t,r))+\beta h(t,r)} \right)^pf^{1-p}(t,r,l), \end{equation}$$ (55) where $$\begin{equation} J\left( {t,r} \right)=-C\int_t^T{e^{A_5 \left( s \right)+A_6 \left( s \right)r}\mathrm{d}s}, \quad J_r =\frac{\partial J\left( {t,r} \right)}{\partial r}=-C\int_t^T{A_6 \left( s \right)e^{A_5 \left( s \right)+A_6 \left( s \right)r}\mathrm{d}s}. \end{equation}$$ (56) Proof. Considering |$g\left ( {t,r,l,z} \right )=x$| and (24), we can get $$\begin{equation} z=\left( {\frac{q}{1-p}(x-J(t,r))+\beta h(t,r)} \right)^{p-1}f^{1-p}(t,r,l). \end{equation}$$ (57) Note that |$H_x =z$|⁠, by integration, we can derive (55). The proof is completed. Remark 3 Theorem 1 displays that one can calculate the optimal proportions invested in the stock and rolling bond by using (53) and (54). In addition, he can also calculate the maximal expected utility of terminal wealth by using (55), when his risk preference meets the HARA utility. Remark 4 From (53) and (54), we can find out some important conclusions as follows: (i) the structures of |$\pi _s^\ast \left ( t \right )$| and |$\pi _B^\ast \left ( t \right )$| are obviously different from those of Guan & Liang (2015); (ii) |$\pi _s^\ast \left ( t \right )$| and |$\pi _B^\ast \left ( t \right )$| are not boundary deterministic functions of time but depend on the random dynamics |$r(t)$|⁠, |$L(t)$| and |$X(t)$|⁠; (iii) |$\pi _s^\ast \left ( t \right )$| depends on all the market parameters |$a,b,k_1,k_2 ,\sigma _r,\lambda _r,\sigma _s,$||$\alpha ,\delta ,\sigma _l,\rho _{sl} , p,q,\beta $| except |$\sigma _r $|⁠, while |$\pi _B^\ast \left ( t \right )$| depends on all the market parameters. 5. Some special cases The HARA utility can be degenerated to a power utility, a logarithm utility and an exponential utility when parameters vary. In this section, we derive the optimal strategies under these special utilities. Special case 1. It is well known that the HARA utility is reduced to a power utility |$U_{pow} (x)={x^p}/p$| when |$\beta =0$| and |$q=1-p$|⁠. So according to Theorem 1 the optimal portfolio under the power utility is immediately obtained as follows: $$\begin{align} \hskip70pt&\pi _{s1}^\ast \left( t \right)=\left( {\frac{1}{1-p}\cdot \frac{L(t)}{X(t)\sigma _s} +\frac{\rho _{sl} \sigma _l} {X(t)\sigma _s }\left( {A_3 \left( t \right)+2A_4 \left( t \right)L(t)} \right)} \right)\left( {X(t)-J(t,r)} \right), \end{align}$$ (58) $$\begin{align}& \pi _{B1}^\ast \left( t \right)=-\frac{1}{1-p}\cdot \frac{\sigma _r L(t)-\lambda _r \sigma _s }{\sigma _s h_0 \left( K \right)X(t)}\left( {X(t)-J(t,r)} \right)\notag \\ &~~~~~~~~~~~~~~ -\frac{1}{X(t)h_0 \left( K \right)}\left( {\left( {X(t)-J(t,r)} \right)A_2 \left( t \right)-J_r } \right)\notag \\ &~~~~~~~~~~~~~~ -\frac{\rho _{sl} \sigma _l \sigma _r }{\sigma _s h_0 \left( K \right)X(t)}\left( {A_3 \left( t \right)+2A_4 \left( t \right)L(t)} \right)\left( {X(t)-J(t,r)} \right), \end{align}$$ (59) and the optimal value function is given by $$\begin{equation} H_{pow}^\ast (t,r,l,x)=\frac{1}{p}\left( {x-J(t,r)} \right)^pf^{1-p}(t,r,l). \end{equation}$$ (60) Special case 2. If we choose |$\beta =0$|⁠, |$p\to 0$| and |$q\to 1$| in (11), we have |$U_{\log } (x)=\ln x$|⁠. Moreover, we get |$A_1 (t)=A_2 (t)=A_3 (t)=A_4 (t)=0$|⁠, which leads to |$f(t,r,l)=1$|⁠. So the optimal portfolio of the problem (10) can be written as follows: $$\begin{equation} \hskip-100pt\pi _{s2}^\ast \left( t \right)=\frac{L(t)}{X(t)\sigma _s} \left( {X(t)-J(t,r)} \right), \end{equation}$$ (61) $$\begin{align} &\pi _{B2}^\ast \left( t \right)=-\frac{\sigma _r L(t)-\lambda _r \sigma _s }{\sigma _s h_0 \left( K \right)X(t)}\left( {X(t)-J(t,r)} \right)+\frac{1}{X(t)h_0 \left( K \right)}J_r, \end{align}$$ (62) and the optimal value function is derived as follows: $$\begin{equation} H_{\log} ^\ast (t,r,l,x)=\ln (x-J(t,r)). \end{equation}$$ (63) Special case 3. If we choose |$\beta =1$| and |$p\to -\infty $| in (11), we have |$U_{\exp } (x)=-{e^{-qx}}/q$|⁠. Under these conditions, we find that |$A_1 (t)\to A_5 (t), \quad A_2 (t)\to A_6 (t), \quad A_3 (t)\to 0$| and |$A_4 (t)\to 0$|⁠. So |$f(t,r,l)\to h(t,r)$|⁠. Taking the limit |$p\to -\infty $| for Equation (55), we have $$\begin{equation} \begin{split} H_{\exp} ^\ast (t,r,l,x)&=\mathop{\lim }\limits_{p\to -\infty } H_{HARA}^\ast (t,r,l,x) \\& =\mathop{\lim }\limits_{p\to -\infty } \frac{1-p}{qp}\left( {\frac{q}{1-p}(x-J(t,r))+\beta h(t,r)} \right)^pf^{1-p}(t,r,l) \\& =-\frac{1}{q}\exp \{-q(x-J(t,r))h^{-1}(t,r)\}h(t,r)\cdot \mathop{\lim }\limits_{p\to -\infty } e^{\ln (f^{-1}(t,r,l)h(t,r))^p}. \end{split} \end{equation}$$ (64) Further, we get $$\begin{equation*} \begin{split} \mathop{\lim} \limits_{p\to -\infty } \ln (f^{-1}(t,r,l)h(t,r))^p&=\mathop{\lim }\limits_{p\to -\infty } \frac{\ln (f^{-1}(t,r,l)h(t,r))}{1/p} \\& =\varphi _1 \left( t \right)+\varphi _2 \left( t \right)r+\varphi _3 \left( t \right)l+\varphi _4 \left( t \right)l^2, \end{split} \end{equation*}$$ where $$\begin{align*} \varphi _1 \left( t \right)=&\ \int_t^T{\left( {-0.5\lambda _r^2 k_2 -\lambda _r k_2 A_6 (s)} \right)\mathrm{d}s} +\int_t^T{(a+\lambda _r k_2 +k_2 A_6 (s))\varphi _2 (s)\mathrm{d}s} \\& +\int_t^T{\left( {\alpha \delta \varphi _3 (s)+\sigma _l^2 \varphi _4 (s)} \right)\mathrm{d}s},\\ \varphi _2 \left( t \right)=&\ \int_t^T{(0.5\lambda _r^2 k_1 +\lambda _r k_1 A_6 (s)-1)e^{\int_t^s{(k_1 A_6 (s)+\lambda _r k_1 -b)\mathrm{d}z}}\mathrm{d}s}, \\ \varphi _3 \left( t \right)=&\ -2\alpha \delta\int_t^T{ \varphi _4 (s)e^{-(\rho _{sl} \sigma _l +\alpha )(s-t)}\mathrm{d}s}, \\ \varphi _4 \left( t \right)=&\ -\frac{1}{4(\rho _{sl} \sigma _l +\alpha )}\left( {e^{-2(\rho _{sl} \sigma _l +\alpha )(T-t)}-1} \right). \end{align*}$$ So the optimal value function for the exponential utility is given by $$\begin{equation} \begin{split} H_{\exp} ^\ast (t,r,l,x)=&-\frac{1}{q}\exp \{-q(x-J(t,r))h^{-1}(t,r) \\& +\varphi _1 (t)+A_5 (t)+\left( {\varphi _2 (t)+A_6 (t)} \right)r+\varphi _3 (t)l+\varphi _4 (t)l^2\}. \end{split} \end{equation}$$ (65) Using (65), we obtain $$\begin{equation} \begin{split} &\frac{H_x} {H_{xx}}=-\frac{1}{q}h, \quad \frac{H_{xl}}{H_{xx}}=-\frac{1}{q}(\varphi _3 (t)+2\varphi _4 (t)l)h, \\& \frac{H_{xr}}{H_{xx}}=-J_r -(x-J)A_6 (t)-\frac{1}{q}\varphi _2 (t)h. \end{split} \end{equation}$$ (66) Substituting (65) back into (13) and (14), we obtain the optimal portfolio for the exponential utility as follows: $$\begin{equation} \hskip-22pt\pi _{s3}^\ast \left( t \right)=\frac{L(t)}{q\sigma _s X(t)}h(t,r)+\frac{\rho _{sl} \sigma _l} {q\sigma _s X(t)}(\varphi _3 (t)+2\varphi _4 (t)L(t))h(t,r), \end{equation}$$ (67) $$\begin{align}& \pi _{B3}^\ast \left( t \right)=-\frac{1}{h_0 \left( K \right)X(t)}\left( {(X(t)-J(t,r))A_6 (t)+\frac{1}{q}\varphi _2 (t)h(t,r)+J_r } \right)\notag \\ &~~~~~~~~~~~~~~~ -\frac{1}{q}\cdot \frac{\rho _{sl} \sigma _l \sigma _r }{\sigma _s h_0 \left( K \right)X(t)}(\varphi _3 (t)+2\varphi _4 (t)L(t))h(t,r)\notag \\ &~~~~~~~~~~~~~~~ -\frac{1}{q}\cdot \frac{\sigma _r L(t)-\lambda _r \sigma _s }{\sigma _s h_0 \left( K \right)X(t)}h(t,r). \end{align}$$ (68) Remark 5 In special cases 1–3, we derive the optimal portfolios for the power utility, the logarithm utility and the exponential utility. Using these results, we can determine the optimal proportions according to different risk preferences. 6. Numerical simulation This section provides a numerical example to investigate the impact of main market parameters on the optimal strategies. For convenience, without loss of generality, the related parameters are supposed as follows: interest rate parameters |$a=0.005, b=0.02, k_1 =0.0016, k_2=0, r(0)=0.05$|⁠; stochastic return parameters |$\alpha =0.01, \delta =0.2, \sigma _l =0.03, \rho _{sl} =0.5, L(0)=0.3$|⁠; stock price parameters |$\sigma _r =0.03, \lambda _r =1, \sigma _s =0.2$|⁠; HARA utility parameters |$p=-2, q=2, \beta =1$|⁠; other parameters are given by |$K=20, T=40, C=0.15, x_0 =1$|⁠. After simulating one track with respect to the interest rate process |$r(t)$|⁠, the stochastic return process |$L(t)$| and the wealth process |$X(t)$|⁠, we get the dynamic evolution processes of the optimal investment strategies over time and some comparisons among these evolution processes, as shown in Fig. 1. Some important conclusions can be drawn from Fig. 1 as follows. (i) At the initial time of investment, the proportion of investment in the bond is negative, while the proportions of investment in the cash and stock are positive, which means the fund manager needs to sell more bonds short. As time passes by, |$\pi _0^\ast (t)$| and |$\pi _s^\ast (t)$| decrease and yet |$\pi _B^\ast (t)$| increases. To be precise, the closer fund members get to retirement age, the smaller the proportion of bonds sold short and the smaller the proportions invested in the cash and stock. (ii) From the point of view of one track simulation, the interest rate keeps fluctuating in the entire time period. To be precise, the interest rate keeps fluctuating around |$0.08$| when |$t \in [0, 30]$|⁠. Then, it starts to go up and arrives at the maximum at |$t=35$|⁠, the maximum is about |$0.13$|⁠. Finally, it fluctuates at about |$0.13$|⁠. (iii) The return process |$L(t)$| keeps fluctuating at the initial value |$0.3$|⁠, which means that the return process is mean-reverting. (iv) The wealth process |$X(t)$| has been on the rise. To be precise, it is growing exponentially. Fig. 1. Open in new tabDownload slide Dynamic evolution processes of the optimal strategies, |$r(t)$|⁠, |$L(t)$| and |$X(t)$| over time. Fig. 1. Open in new tabDownload slide Dynamic evolution processes of the optimal strategies, |$r(t)$|⁠, |$L(t)$| and |$X(t)$| over time. Because |$\pi _s^\ast \left ( t \right )$| and |$\pi _B^\ast \left ( t \right )$| depend on the random processes |$r(t)$|⁠, |$L(t)$| and |$X(t)$|⁠, we use Monte Carlo simulation technique to analyse the dynamic effect of parameters on the optimal strategies. We simulate |$10000$| tracks of |$r(t)$|⁠, |$L(t)$| and |$X(t)$| with respect to time and calculate the average of the |$10000$| tracks. By the technique of simulation, we give the following Figs 2–4. Fig. 2. Open in new tabDownload slide Dynamic effect of |$b$| and |$k_1$| on |$\pi _B^\ast (t)$| and |$\pi _s^\ast (t)$| over time. Fig. 2. Open in new tabDownload slide Dynamic effect of |$b$| and |$k_1$| on |$\pi _B^\ast (t)$| and |$\pi _s^\ast (t)$| over time. Fig. 3. Open in new tabDownload slide Dynamic effect of |$\alpha $| and |$\sigma _l$| on |$\pi _B^\ast (t)$| and |$\pi _s^\ast (t)$| over time. Fig. 3. Open in new tabDownload slide Dynamic effect of |$\alpha $| and |$\sigma _l$| on |$\pi _B^\ast (t)$| and |$\pi _s^\ast (t)$| over time. Fig. 4. Open in new tabDownload slide Dynamic effect of |$C$| and |$\rho _{sl}$| on |$\pi _B^\ast (t)$| and |$\pi _s^\ast (t)$| over time. Fig. 4. Open in new tabDownload slide Dynamic effect of |$C$| and |$\rho _{sl}$| on |$\pi _B^\ast (t)$| and |$\pi _s^\ast (t)$| over time. Figure 2 shows the dynamic effect of interest rate parameters |$b$| and |$k_1$| on |$\pi _s^\ast (t)$| and |$\pi _B^\ast (t)$|⁠. We can draw from Fig. 2 that |$\pi _B^\ast (t)$| decreases in |$b$| and |$k_1$| for any given time while |$\pi _s^\ast (t)$| increases in |$b$| and |$k_1$| for any given time. From the economic implications of |$b$|⁠, the larger the value of |$b$|⁠, the smaller the expected value of interest rate, which leads to the smaller yield of the bond. As a result, the fund manager will invest less proportion in the rolling bond and more proportion in the stock. On the other hand, |$k_1$| determines the size of volatility resulted from the interest rate dynamics. When the value of |$k_1$| increases, the volatility of interest rate increases as well, which leads to larger risk of the rolling bond, so the fund manager needs to reduce the proportion invested in the rolling bond and add the proportion in the stock. Finally, we find out that the effect of |$b$| on |$\pi _s^\ast (t)$| and |$\pi _B^\ast (t)$| is larger than the effect of |$k_1$|⁠. Figure 3 displays the dynamic effect of stochastic return parameters |$\alpha $| and |$\sigma _l $| on |$\pi _s^\ast (t)$| and |$\pi _B^\ast (t)$|⁠. From Fig. 3, we can see the following conclusions. (i) |$\pi _B^\ast (t)$| and |$\pi _s^\ast (t)$| decrease with respect to |$\alpha $| for any given time. As a matter of fact, the expected return of the stock is decreasing as the value of |$\alpha $| is increasing. As a result, the lower yield makes the fund manager invest less money in the stock. (ii) |$\pi _s^\ast (t)$| decreases with respect to |$\sigma _l $| for any given time as well and the effect of |$\sigma _l $| on |$\pi _B^\ast (t)$| is negligible. The larger the value of |$\sigma _l $|⁠, the larger the volatility of stochastic return. It will lead to that the volatility in the stock market increases because of the positive correlation between stock market and stochastic return. So the optimal proportion invested in the stock will descend. (iii) |$\pi _s^\ast (t)$| with larger |$\alpha $| presents the rather weird behaviour, which is different from those with lower values of |$\alpha $|⁠. From the view of point of investor’s behaviour, a larger value of |$\alpha $| leads to a lower expected return of the stock; it means that the stock market is bear, so the fund manager invests less money in the stock at the initial time. As time goes on, pension funds accumulate more and more. In order to maintain and increase the value of pension fund, the fund manager needs to put more money into the stock market, so the proportion of investment in the stock will gradually increase. However, in the years leading up to retirement, the fund manager needs to gradually reduce the proportion of investment in the stock in order to ensure that the pensions of members are paid in full and on time. Therefore, the trend shown in Fig. 3 is close to the reality. Figure 4 analyzes the dynamic effect of |$\rho _{sl} $| and |$C$| on |$\pi _s^\ast (t)$| and |$\pi _B^\ast (t)$|⁠. Some important conclusions can be seen from Fig. 4 as follows. (i) |$\pi _s^\ast (t)$| will increase while |$\pi _B^\ast (t)$| will decrease for any given time as the contribution rate |$C$| is increasing. It signifies that the fund manager should hold more proportion in the stock and simultaneously hold less proportion in the rolling bond. As a matter of fact, this conclusion is consistent with our intuition. The larger the value of |$C$|⁠, the more the wealth of fund manager. Thus, it is very necessary for a fund manager to invest more wealth into the stock market in order to preserve and increase the value. (ii) When the correlation |$\rho _{sl}$| is positive, the greater the positive correlation, the greater the proportion of money invested in the stock for any given time; when |$\rho _{sl}$| is negative, the smaller the value of |$\rho _{sl}$|⁠, namely, the greater the negative correlation, the greater the proportion of money invested in stocks as well. To sum up, |$\pi _s^\ast (t)$| is an increasing function with respect to |$\|\rho _{sl}\|$|⁠. In addition, we find out that the optimal proportion with negative correlation invested in the stock is more than that with positive correlation for the same |$\|\rho _{sl}\|$|⁠. (iii) The value of |$\rho _{sl}$| has less effect on |$\pi _B^\ast (t)$|⁠. 7. Conclusions This paper studies a DC pension management problem in a general utility framework, where the utility function is assumed to be a HARA utility including the power utility, the exponential utility and the logarithmic utility as special cases. Due to the complexity of the optimal investment problems with HARA utility in an incomplete financial market, we introduce Legendre transform-dual theory to change the nonlinear HJB equation into its linear dual one. By using variable change method, we obtain the closed-form solution of the optimal portfolio. In addition, some special cases are systematically derived. A numerical example is provided to illustrate our results. Our research displays that the Legendre transform-dual theory is an effective method in dealing with some more sophisticated portfolio selection problems with HARA preference. In addition, applications on Legendre transform-dual theory have some important advantages: (i) it can transform a nonlinear HJB equation into a linear dual one; (ii) the boundary condition of the dual equation under the HARA utility is a linear structure, which makes the structure of solution of the dual equation be conjectured more easily; (iii) compared with the martingale method or the duality method, the solving process is more convenient and understandable; (iv) the difficulties in the application process are more easily overcome such that the Legendre transform-dual theory can be used to solve more problems, which is the most important reason that the Legendre transform-dual theory is widely used to tackle some more sophisticated portfolio selection problems; (v) the explicit solution to the original HJB equation is obtained by using Legendre transform-dual theory, which lays a solid foundation for directly conjecturing the structure of solution of the HJB equation. Numerical results reveal some important conclusions: (i) the contribution rate |$C$| and interest rate parameter |$b$| have considerable effect on |$\pi _B^\ast (t)$| and |$\pi _s^\ast (t)$|⁠; (ii) the correlation coefficient |$\rho _{sl}$| has considerable effect on |$\pi _s^\ast (t)$|⁠; (iii) the effect of |$\sigma _l $| and |$\rho _{sl}$| on |$\pi _B^\ast (t)$| is negligible; (iv) the dynamic behaviour of |$\pi _s^\ast (t)$| with larger |$\alpha $| is different from those with lower values of |$\alpha $|⁠. According to our research, the method used in this paper is generic and can also be used to solve other more sophisticated portfolio selection problems, for example, the DC pension plans with stochastic wage or with minimum guarantee or with the return of premiums clauses. However, there are some challenges in the application of Legendre transform-dual theory. The greatest challenge lies in solving those subsequent partial differential equations, which is the key whether the closed-form solutions of optimal portfolios can be obtained successfully. Meanwhile, pension fund management problems are getting more and more appealing and have greatly attracted more attentions. To study and solve pension fund management problems with other more practical environments has important academic values and wide application prospects. Acknowledgements The authors are very grateful to reviewers for their comments and suggestions, which greatly improve the quality of this paper. Funding National Natural Science Foundation of China (71671122); Humanities and Social Science Research Foundation of Ministry of Education of China (16YJA790004); Tianjin’s University ‘Youth Backbone Innovation Talent Training Program’ Funded Project; China Postdoctoral Science Foundation Funded Project (2016T90203). References Blake , D. , Wright , D. & Zhang , Y. 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TI - Defined contribution pension planning with a stochastic interest rate and mean-reverting returns under the hyperbolic absolute risk aversion preference
JO - IMA Journal of Management Mathematics
DO - 10.1093/imaman/dpz009
DA - 2020-02-28
UR - https://www.deepdyve.com/lp/oxford-university-press/defined-contribution-pension-planning-with-a-stochastic-interest-rate-1e2CE107YL
SP - 167
VL - 31
IS - 2
DP - DeepDyve
ER -