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 (Omega, F, {F}(t >= 0), P) where the coefficients are adapted to a filtration G = {g(t)}(0 <= t <= T), with F-t subset of g(t) for all t is an element of [0, T], T > 0 being a fixed terminal time. By an insider in this market we mean a person who has access to a filtration (information) H = {H-t}(0 <= t <= T) which is strictly bigger than the filtration G = {G(t)}(0 <= t <= 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 superset of 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 R, This is a generalization of results in [16], [20] and [2].