Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations with Jumps

TL;DR

This paper develops a stochastic viscosity solution framework for fully nonlinear jump-diffusion Hamilton--Jacobi--Bellman equations, establishing existence, uniqueness, and the dynamic programming principle for stochastic control problems with jumps.

Contribution

It introduces a novel stochastic viscosity solution approach for non-local HJB equations with jumps, handling polynomial growth and establishing key theoretical properties.

Findings

Established the dynamic programming principle for jump-diffusion control.

Proved existence of solutions using measurable selection and Itô--Kunita formula.

Proved global uniqueness under super-parabolicity condition.

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.