On the Approximation of Differential Equations Driven by Some Random Processes as Rough Paths

TL;DR

This paper investigates the approximation of stochastic differential equations driven by certain random processes using rough path theory, focusing on limit behavior and convergence in a singular perturbation context.

Contribution

It introduces a method to lift random processes as rough paths and applies averaging and convergence techniques to analyze their limits in stochastic differential equations.

Findings

Established moment estimates for the random process and its lift.

Applied rough path topology to prove convergence of the approximations.

Identified the limit of the stochastic differential equations driven by the processes.

Abstract

We explore the limit of stochastic differential equations driven by some random processes satisfying singularly perturbed second order stochastic differential equations. The main tool we employ is the universal limit theorem in rough path theory. To this end, we lift the random process as a rough path in a natural manner. After suitable change-of-variable, the random process has a form of slow-fast system. Moment estimates of both the random process and its lift are given, followed by which, averaging technique and convergence theorem in rough path topology are used to identify the limit.