The averaging principle for stochastic differential equations driven by a Wiener process revisited

TL;DR

This paper revisits the averaging principle for one-dimensional stochastic differential equations driven by Wiener processes, emphasizing the importance of averaging the square of the diffusion coefficient and decomposing the solution into average and fluctuation components.

Contribution

It introduces a refined averaging principle that separates the solution into average and fluctuation parts, clarifying the role of the squared diffusion coefficient in the limit behavior.

Findings

The solution decomposes into average and fluctuation terms.

Both terms influence the limit, highlighting the necessity of averaging the square of the diffusion coefficient.

The approach provides a clearer understanding of the stochastic averaging process.

Abstract

We consider a one-dimensional stochastic differential equation driven by a Wiener process, where the diffusion coefficient depends on an ergodic fast process. The averaging principle is satisfied: it is well-known that the slow component converges in distribution to the solution of an averaged equation, with generator determined by averaging the square of the diffusion coefficient. We propose a version of the averaging principle, where the solution is interpreted as the sum of two terms: one depending on the average of the diffusion coefficient, the other giving fluctuations around that average. Both the average and fluctuation terms contribute to the limit, which illustrates why it is required to average the square of the diffusion coefficient to find the limit behavior.