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