Stochastic Perron for Stochastic Target Problems

TL;DR

This paper extends stochastic Perron's method to analyze stochastic target problems with unbounded controls and jumps, establishing viscosity solutions and generalizing classical control results to jump-diffusion settings.

Contribution

It adapts stochastic Perron's method to jump-diffusion target problems with unbounded controls, proving uniqueness of viscosity solutions and extending classical control results.

Findings

Constructed viscosity sub- and super-solutions for HJB equations.

Proved uniqueness of viscosity solutions under comparison principles.

Generalized classical control results to jump-diffusion scenarios.

Abstract

In this paper, we adapt stochastic Perron's method to analyze a stochastic target problem with unbounded controls in a jump diffusion set-up. With this method, we construct a viscosity sub-solution and super-solution to the associated Hamiltonian-Jacobi-Bellman (HJB) equations. Under comparison principles, uniqueness of the viscosity solutions holds and the value function coincides with the unique solution in the parabolic interior. Since classical control problems can be analyzed under the framework of stochastic target problems (with unbounded controls), we use our results to generalize the results in ArXiv:1212.2170 to problems with controlled jumps.