Discrete approximation to solution flows of Tanaka SDE related to Walsh Brownian motion

TL;DR

This paper demonstrates how solutions to a specific Tanaka SDE related to Walsh Brownian motion can be approximated through discrete models, establishing a connection between discrete and continuous stochastic processes.

Contribution

It introduces a discrete approximation scheme for the unique solutions of a Walsh Brownian motion-related Tanaka SDE, bridging discrete models with continuous stochastic flows.

Findings

Discrete models converge to the unique Wiener solution.

Discrete approximations also converge to the flow of mappings.

The approach clarifies the relationship between discrete and continuous solutions.

Abstract

In a previous work, we have defined a Tanaka SDE related to Walsh Brownian motion which depends on kernels. It was shown that there are only one Wiener solution and only one flow of mappings solving this equation. In the terminology of Le Jan and Raimond, these are respectively the stronger and the weaker among all solutions. In this paper, we obtain these solutions as limits of discrete models.