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arXiv · 2609.13828

Viscosity Solutions for Singular HJB Equations: BSDE Representations and Stochastic Control

Dirk Becherer·Nicolas Perkowski·Yuchen Sun·Carlos Villanueva Mariz
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Abstract

We introduce a notion of viscosity solution for Hamilton--Jacobi--Bellman (HJB) equations with distributional drift, based on paracontrolled test functions and related through a Zvonkin transformation to classical viscosity theory. The equations considered are of the form \[ \left(\partial_t+\frac12Δ+b\cdot\nabla\right)h(t,x) =-H(t,x,h(t,x),\nabla h(t,x)), \] where $b$ is singular in the sense of \cite{paradistrib} and has regularity $\mathcal C^{-α}$ for $α\in(1/2,2/3)$. Using doubling-of-variables arguments, we derive a priori gradient estimates that also cover Hamiltonians with slightly superquadratic growth in $\nabla h$. We also obtain probabilistic representations through singular control problems for convex $H$ and weak singular forward--backward SDEs for possibly nonconvex Hamiltonians with at most quadratic growth.

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Dirk Becherer, Nicolas Perkowski, Yuchen Sun, Carlos Villanueva Mariz. 2026-09-12. Viscosity Solutions for Singular HJB Equations: BSDE Representations and Stochastic Control. https://arxiv.org/abs/2609.13828

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