Moderate Averaged Deviations for a Multi-Scale System with Jumps and Memory

Abstract

This work studies a two-time-scale functional system given by two jump diffusions under the scale separation by a small parameter ε→0. The coefficients of the equations that govern the dynamics of the system depend on the segment process of the slow variable (responsible for capturing delay effects on the slow component) and on the state of the fast variable. We derive a moderate deviation principle for the slow component of the system in the small noise limit using the weak convergence approach. The rate function is written in terms of the averaged dynamics associated with the multi-scale system. The core of the proof of the moderate deviation principle is the establishment of an averaging principle for the auxiliary controlled processes associated with the slow variable in the framework of the weak convergence approach. The controlled version of the averaging principle for the jump multi-scale diffusion relies on a discretization method inspired by the classical Khasminkii’s averaging principle.

Keywords

• moderate deviation principle
• multi-scale stochastic differential equations with jumps and delay
• segment process
• stochastic averaging principle
• weak convergence approach