Maximum principle for stochastic control of SDEs with measurable drifts

TL;DR

This paper develops a stochastic maximum principle for controlling systems governed by SDEs with irregular, measurable drifts, using approximation and local time techniques to handle non-smooth coefficients.

Contribution

It introduces a novel maximum principle for SDEs with irregular drifts, employing Sobolev derivatives and local time representations, advancing stochastic control theory.

Findings

Established a necessary and sufficient maximum principle.

Derived explicit Sobolev-based variation representations.

Applied approximation and Ekeland's principle for non-smooth drifts.

Abstract

In this paper, we consider stochastic optimal control of systems driven by stochastic differential equations with irregular drift coefficient. We establish a necessary and sufficient stochastic maximum principle. To achieve this, we first derive an explicit representation of the first variation process (in Sobolev sense ) of the controlled diffusion. Since the drift coefficient is not smooth, the representation is given in terms of the local time of the state process. Then we construct a sequence of optimal control problems with smooth coefficients by an approximation argument. Finally, we use Ekeland's variational principle to obtain an approximating adjoint process from which we derive the maximum principle by passing to the limit.