Viscosity Solutions for Singular HJB Equations: BSDE Representations and Stochastic Control
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 (∂t+1/2Δ+b·∇)h(t,x) =−H(t,x,h(t,x),∇ h(t,x)), where b is singular in the sense of \cite{paradistrib} and has regularity 𝒞−α for α∈(1/2,2/3). Using doubling-of-variables arguments, we derive a priori gradient estimates that also cover Hamiltonians with slightly superquadratic growth in ∇ 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.
Related
- Existence and Uniqueness of Viscosity Solutions for Stochastic Hamilton--Jacobi--Bellman Equations
- Singular viscosity-exponent limit for one-dimensional compressible Navier-Stokes equations with far field vacuum
- Aleksandrov-Bakelman-Pucci maximum principle for Ln-viscosity solutions of uniformly elliptic equations with unbounded terms: The critical case
- Pathological viscosity solutions of Hamilton--Jacobi equations
- Viscosity Solutions for Hamilton-Jacobi equations on Separable Banach Spaces
BibTeX
@misc{becherer2026viscosity,
title = {{Viscosity Solutions for Singular HJB Equations: BSDE Representations and Stochastic Control}},
author = {Dirk Becherer and Nicolas Perkowski and Yuchen Sun and Carlos Villanueva Mariz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13828},
eprint = {2609.13828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13828},
}
RIS
TY - UNPB
TI - Viscosity Solutions for Singular HJB Equations: BSDE Representations and Stochastic Control
AU - Dirk Becherer
AU - Nicolas Perkowski
AU - Yuchen Sun
AU - Carlos Villanueva Mariz
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13828
UR - https://arxiv.org/abs/2609.13828
AB - 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 (∂_t+1/2Δ+b·∇)h(t,x) =−H(t,x,h(t,x),∇ h(t,x)), where b is singular in the sense of \cite{paradistrib} and has regularity 𝒞^−α for α∈(1/2,2/3). Using doubling-of-variables arguments, we derive a priori gradient estimates that also cover Hamiltonians with slightly superquadratic growth in ∇ 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.
ER -