Verification Theorems for HJB Equations with α-Stable Lévy Noise
A Viscosity-Solution Framework for Non-Local Generators
Abstract
We establish verification theorems for finite-horizon Hamilton–Jacobi–Bellman equations driven by $\alpha$-stable Lévy noise, where the classical Itô 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 numerical scheme based on jump-adapted Monte Carlo is provided and converges at rate $O(N^{-1/\alpha})$ in the empirical norm.