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