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arXiv · 2604.12153

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

Akan Selim·Siddhartha Ganguly·Ali Pakniyat·Panagiotis Tsiotras
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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.

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Akan Selim, Siddhartha Ganguly, Ali Pakniyat, Panagiotis Tsiotras. 2026-04-14. Nonlinear Stochastic Optimal Control and Optimal Stopping using the Fokker-Planck Transformation. https://arxiv.org/abs/2604.12153

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