Partial observation control in an anticipating environment

A study is made of a controlled stochastic system whose state $X(t)$ at time $t$ is described by a stochastic differential equation driven by Lévy processes with filtration $\{\mathscr F_t\}_{t\in[0,T]}$. The system is assumed to be anticipating, in the sense that the coefficients are assumed to be adapted to a filtration $\{\mathscr G_t\}_{t\geqslant0}$ with $\mathscr F_t\subseteq\mathscr G_t$ for all $t\in[0,T]$. The corresponding anticipating stochastic differential equation is interpreted in the sense of forward integrals, which naturally generalize semimartingale integrals. The admissible controls are assumed to be adapted to a filtration $\{\mathscr E_t\}_{t\in[0,T]}$ such that $\mathscr E_t\subseteq\mathscr F_t$ for all $t\in[0,T]$. The general problem is to maximize a given performance functional of this system over all admissible controls. This is a partial observation stochastic control problem in an anticipating environment. Examples of applications include stochastic volatity models in finance, insider influenced financial markets, and stochastic control of systems with delayed noise effects. Some particular cases in finance, involving optimal portfolios with logarithmic utility, are solved explicitly.