Stochastic maximum principle for systems driven by local martingales with spatial parameters

TL;DR

This paper develops a stochastic maximum principle for control systems driven by local martingales with spatial parameters, providing necessary and sufficient conditions for optimality and discussing linear quadratic cases.

Contribution

It introduces a maximum principle for systems driven by local martingales with spatial parameters, extending stochastic control theory.

Findings

Derived necessary conditions for optimal controls.

Proved sufficiency under certain conditions.

Analyzed stochastic linear quadratic problems.

Abstract

We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter. Assuming the convexity of the control domain, we obtain the stochastic maximum principle as the necessary condition for an optimal control, and we also prove its sufficiency under proper conditions. The stochastic linear quadratic problem in this setting is also discussed.