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 (SDEs) with irregular drift. The drift coefficient is assumed to decompose as [Formula: see text], where [Formula: see text] is bounded and Borel measurable, [Formula: see text] has bounded variation, and [Formula: see text] 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.