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 $displaystyle (Omega,F,left{Fright}_{tgeq 0},P)$ where the coefficients are adapted to a filtration ${Bbb G}=left{G_tright}_{0leq tleq T}$, with $F_tsubsetG_t$ for all $tin [0,T]$, $T>0$ being a fixed terminal time. By an {it insider} in this market we mean a person who has access to a filtration (information) $displaystyle{Bbb H}=left{H_tright}_{0leq tleq T}$ which is strictly bigger than the filtration $displaystyle{Bbb G}=left{G_tright}_{0leq tleq T}$. 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 $H_t supsetG_t$ 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].