An Anticipative Stochastic Minimum Principle under Enlarged Filtrations

Abstract

We prove an anticipative sufficient stochastic minimum principle in a jump process setup with initially enlarged filtrations. We apply the result to several portfolio selection problems like mean and minimal variance hedging under enlarged filtrations. We also investigate utility maximizing portfolio selection under future information. Contrarily to classical optimization methods like dynamic programming, our stochastic minimum principle likewise applies to non-Markovian setups. On the mathematical side, we are concerned with jump processes, forward and backward stochastic differential equations and forward integrals.

Keywords: stochastic minimum principle, optimal control, mean/minimal variance hedging, wealth process, self-financing portfolio, anticipative calculus, forward integral, forward stochastic differential equation, enlargement of filtration, future information, insider trading, Lévy process

JEL Classification: G11, G12, G14, C02, C61