Nonlinear Stochastic Optimal Control and Optimal Stopping using the Fokker-Planck Transformation

TL;DR

This paper introduces a density-based deterministic framework for nonlinear stochastic optimal control with stopping, connecting Fokker-Planck equations to Hamilton-Jacobi-Bellman formulations.

Contribution

It develops a novel density-based representation for stochastic control problems, enabling new analytical conditions for optimal stopping and control.

Findings

Reformulates controlled Fokker-Planck as a continuity equation with a velocity field.

Shows the distributional dynamic programming equation shares the same differential operator as HJB.

Derives first-order necessary conditions for optimal control with state-dependent stopping.

Abstract

In this paper, we develop a theoretical framework for nonlinear stochastic optimal control problems with optimal stopping by establishing a density-based deterministic representation of the underlying diffusion. For state-independent diffusion, we rewrite the controlled Fokker-Planck equation as a continuity equation driven by a score-corrected velocity field, yielding a deterministic characteristic dynamics that reproduces the marginal law of the stochastic system. Leveraging Stein-type identities, we show that the associated distributional dynamic programming equation admits the same second-order differential operator as the distributional stochastic Hamilton-Jacobi-Bellman formulation. Building on this representation, we formulate an optimal control problem with state-dependent terminal-time assignment and terminal distributional constraints and derive the first-order necessary conditions using variational analysis. We present the conditions both for a common terminal time and for the general case of state-dependent stopping.