Viscosity methods for large deviations estimates for multiscale stochastic processes

TL;DR

This paper develops viscosity methods to derive large deviations estimates for multiscale stochastic processes, particularly focusing on short maturity asymptotics in systems with stochastic volatility modeled by fast-evolving processes.

Contribution

It introduces viscosity solution techniques for singular perturbation problems in second order HJB equations within unbounded domains, applicable to stochastic volatility models.

Findings

Established large deviations estimates for multiscale stochastic systems.

Provided a framework for short maturity asymptotics in stochastic volatility models.

Extended viscosity methods to unbounded settings for complex stochastic processes.

Abstract

We study singular perturbation problems for second order HJB equations in an unbounded setting. The main applications are large deviations estimates for the short maturity asymptotics of stochastic systems affected by a stochastic volatility, where the volatility is modelled by a process evolving at a faster time scale and satisfying some condition implying ergodicity.