Recurrent Solutions of Stochastic Differential Equations with Non-constant Diffusion Coefficients which obey the Law of the Iterated Logarithm

TL;DR

This paper investigates a class of stochastic differential equations with non-constant diffusion coefficients, establishing conditions under which their solutions exhibit behavior similar to Brownian motion, including adherence to the Law of the Iterated Logarithm.

Contribution

It introduces a method using scale and space transformations to analyze SDE solutions with non-constant diffusion, extending understanding of their asymptotic behavior.

Findings

Solutions obey the Law of the Iterated Logarithm under certain conditions

Provides sufficient criteria for LIL behavior in non-constant diffusion SDEs

Extends classical Brownian motion results to more general stochastic processes

Abstract

By using a change of scale and space, we study a class of stochastic differential equations (SDEs) whose solutions are drift--perturbed and exhibit behaviour analogous to standard Brownian motion including to the Law of the Iterated Logarithm (LIL). Sufficient conditions ensuring that these processes obey the LIL are given.