Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations with Jumps
Abstract
This paper studies the stochastic optimal control of jump-diffusion processes and the associated fully nonlinear backward stochastic Hamilton--Jacobi--Bellman (BSHJB) equations. We establish the dynamic programming principle (DPP) via backward semigroups to characterize the value function. To handle non-local integro-differential operators and polynomial growth, we introduce a stochastic viscosity solution framework based on semimartingale test functions and global tangency conditions. Existence is proved using the measurable selection theorem and the generalized It\^o--Kunita formula. Finally, under a super-parabolicity condition, we establish a weak comparison principle and prove global uniqueness via localized bounding envelopes and backward induction.
Keywords
Cite
@article{arxiv.2605.20593,
title = {Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations with Jumps},
author = {Dunxiang Liang and Qingxin Meng},
journal= {arXiv preprint arXiv:2605.20593},
year = {2026}
}