# A General Stochastic Calculus Approach to Insider Trading

A General Stochastic Calculus Approach to Insider Trading The purpose of this paper is to present a general stochastic calculus approach to insider trading. We consider a market driven by a standard Brownian motion $B(t)$ on a filtered probability space $\displaystyle (\Omega,\F,\left\{\F\right\}_{t\geq 0},P)$ where the coefficients are adapted to a filtration ${\Bbb G}=\left\{\G_t\right\}_{0\leq t\leq T}$, with $\F_t\subset\G_t$ for all $t\in (0,T)$, $T>0$ being a fixed terminal time. By an {\it insider} in this market we mean a person who has access to a filtration (information) $\displaystyle{\Bbb H}=\left\{\H_t\right\}_{0\leq t\leq T}$ which is strictly bigger than the filtration $\displaystyle{\Bbb G}=\left\{\G_t\right\}_{0\leq t\leq T}$. In this context an insider strategy is represented by an $\H_t$-adapted process $\phi(t)$ and we interpret all anticipating integrals as the forward integral defined in (23) and (25). We consider an optimal portfolio problem with general utility for an insider with access to a general information $\H_t \supset\G_t$ and show that if an optimal insider portfolio $\pi^*(t)$ of this problem exists, then $B(t)$ is an $\H_t$-semimartingale, i.e. the enlargement of filtration property holds. This is a converse of previously known results in this field. Moreover, if $\pi^*$ exists we obtain an explicit expression in terms of $\pi^*$ for the semimartingale decomposition of $B(t)$ with respect to $\H_t$. This is a generalization of results in (16), (20) and (2). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# A General Stochastic Calculus Approach to Insider Trading

, Volume 52 (2) – Aug 1, 2005
15 pages

Publisher
Springer-Verlag
Subject
Mathematics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Methods
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-005-0825-2
Publisher site
See Article on Publisher Site

### Abstract

The purpose of this paper is to present a general stochastic calculus approach to insider trading. We consider a market driven by a standard Brownian motion $B(t)$ on a filtered probability space $\displaystyle (\Omega,\F,\left\{\F\right\}_{t\geq 0},P)$ where the coefficients are adapted to a filtration ${\Bbb G}=\left\{\G_t\right\}_{0\leq t\leq T}$, with $\F_t\subset\G_t$ for all $t\in (0,T)$, $T>0$ being a fixed terminal time. By an {\it insider} in this market we mean a person who has access to a filtration (information) $\displaystyle{\Bbb H}=\left\{\H_t\right\}_{0\leq t\leq T}$ which is strictly bigger than the filtration $\displaystyle{\Bbb G}=\left\{\G_t\right\}_{0\leq t\leq T}$. In this context an insider strategy is represented by an $\H_t$-adapted process $\phi(t)$ and we interpret all anticipating integrals as the forward integral defined in (23) and (25). We consider an optimal portfolio problem with general utility for an insider with access to a general information $\H_t \supset\G_t$ and show that if an optimal insider portfolio $\pi^*(t)$ of this problem exists, then $B(t)$ is an $\H_t$-semimartingale, i.e. the enlargement of filtration property holds. This is a converse of previously known results in this field. Moreover, if $\pi^*$ exists we obtain an explicit expression in terms of $\pi^*$ for the semimartingale decomposition of $B(t)$ with respect to $\H_t$. This is a generalization of results in (16), (20) and (2).

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Aug 1, 2005

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