From Reflecting Brownian Motion to Reflected Stochastic Differential Equations: A Systematic Survey and Complementary Study

TL;DR

This paper provides a comprehensive survey of reflecting Brownian motion, explores extensions of the Skorohod problem, and discusses the existence of solutions for reflected stochastic differential equations, including multidimensional cases.

Contribution

It offers a systematic review of reflecting Brownian motion and extends the Skorohod problem to more general settings, including multidimensional convex domains.

Findings

Multidimensional Skorohod equation solvable in convex domains

Extended the solution framework for reflecting Brownian motion

Discussed existence conditions for reflected stochastic differential equations

Abstract

This work contributes a systematic survey and complementary insights of reflecting Brownian motion and its properties. Extension of the Skorohod problem's solution to more general cases is investigated, based on which a discussion is further conducted on the existence of solutions for a few particular kinds of stochastic differential equations with a reflected boundary. It is proved that the multidimensional version of the Skorohod equation can be solved under the assumption of a convex domain (D).