Convergence Implications via Dual Flow Method

TL;DR

This paper studies the approximation of dual stochastic flows associated with one-dimensional SDEs, demonstrating how discrete-time approximations can effectively capture the behavior of reflecting diffusions through weak and strong convergence analysis.

Contribution

It introduces a discrete-time stochastic-flow approximation for dual flows of SDEs and analyzes their convergence to reflecting diffusions.

Findings

Discrete-time dual flows approximate reflecting diffusions.

Weak and strong convergence of the approximations are established.

Provides theoretical insights into stochastic flow duality and convergence.

Abstract

Given a one-dimensional stochastic differential equation, one can associate to this equation a stochastic flow on $[0,+\infty )$, which has an absorbing barrier at zero. Then one can define its dual stochastic flow. In \cite{AW}, Akahori and Watanabe showed that its one-point motion solves a corresponding stochastic differential equation of Skorokhod-type. In this paper, we consider a discrete-time stochastic-flow which approximates the original stochastic flow. We show that under some assumptions, one-point motions of its dual flow also approximates the corresponding reflecting diffusion. We investigate the relation between them in weak and strong approximation sense.