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Applied Mathematics and Optimization

, Volume 52, Issue 2, pp 167–181 | Cite as

A General Stochastic Calculus Approach to Insider Trading

  • Francesca Biagini
  • Bernt Øksendal
Article

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].

Forward integral Skorohod integral Wick product Insider trading Utility function 

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematics, University of Bologna, Piazza di Porta S. Donato 5, I-40127 BolognaItaly
  2. 2.Center of Mathematics for Applications (CMA), Department of Mathematics, University of Oslo, Box 1053 Blindern, N-0316 OsloNorway
  3. 3.Norwegian School of Economics and Business Administration, Helleveien 30, N-5045 BergenNorway

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