Stochastic Optimal Control for Systems with Drifts of Bounded Variation: A Maximum Principle Approach

TL;DR

This paper establishes a stochastic maximum principle for nonlinear systems with irregular drift, using new results on SDEs with bounded variation drifts and an approximation scheme.

Contribution

It introduces a Pontryagin-type maximum principle for systems with minimal regularity assumptions on the drift, including irregular and bounded variation components.

Findings

Proves existence and uniqueness of solutions for SDEs with bounded variation drift.

Provides an explicit representation of the first variation process via local time.

Derives an optimal control policy for an insurance surplus model.

Abstract

We study a stochastic control problem for nonlinear systems governed by stochastic differential equations with irregular drift. The drift coefficient is assumed to decompose as $b(t,x,a)=b_1(t,x)+b_2(x)b_3(t,a)$, where $b_1$ is bounded and Borel measurable, $b_2$ has bounded variation, and $b_3$ is bounded and smooth. Under these minimal regularity assumptions, we establish a Pontryagin-type stochastic maximum principle. The analysis relies on new results for SDEs with random drift of bounded variation, including existence, uniqueness, and Malliavin-Sobolev differentiability of the state process. A key ingredient is an explicit representation of the first variation process obtained via integration with respect to the space-time local time of bounded variation processes. By combining a suitable approximation scheme with Ekeland's variational principle, and using a Garcia-Rodemich-Rumsey inequality to obtain a uniform control of the first variation, we derive the maximum principle. As an application, we derive an optimal corridor-type capital adjustment policy for an insurance surplus model.