Verification Theorems for HJB Equations with α-Stable Lévy Noise

We establish verification theorems for finite-horizon Hamilton–Jacobi–Bellman equations driven by $\alpha$-stable L\’evy noise, where the classical It\^o calculus is replaced by a non-local infinitesimal generator. For $\alpha \in (1, 2)$ and bounded controls, we prove existence and uniqueness of viscosity solutions, derive optimal feedback laws expressible in terms of the resolvent of the non-local operator, and recover the classical Brownian verification theorem as $\alpha \to 2$. A jump-adapted Monte Carlo scheme for the controlled state is given; on a linear–quadratic subcase with a closed-form value function its statistical error decays as $N^{-1/2}$ at every $\alpha$ (fitted slopes $-0.504$ to $-0.489$ for $\alpha \in \{1.3, 1.5, 1.7, 1.9\}$), as plain Monte Carlo must, and its bias is first order in the time step; extrapolated to $\Delta t \to 0$ it reproduces the closed-form value to $0.11$ % ($1.0$ standard errors).