Stochastic Integral Equations for Walsh Semimartingales

TL;DR

This paper develops a framework for Walsh semimartingales, including Walsh Brownian motion, deriving integral equations and change-of-variable formulas, and analyzing their solvability, uniqueness, and connections to martingale problems.

Contribution

It introduces Walsh semimartingales, derives associated stochastic integral equations and formulas, and studies their solvability and uniqueness in Markovian contexts.

Findings

Constructed planar Walsh semimartingales including Walsh Brownian motion.

Derived Harrison-Shepp-type equations and change-of-variable formulas.

Analyzed solvability and uniqueness of the resulting stochastic integral systems.

Abstract

We construct planar semimartingales that include the Walsh Brownian motion as a special case, and derive Harrison-Shepp-type equations and a change-of-variable formula in the spirit of Freidlin-Sheu for these so-called "Walsh semimartingales". We examine the solvability of the resulting system of stochastic integral equations. In appropriate Markovian settings we study two types of connections to martingale problems, questions of uniqueness in distribution for such processes, and a few examples.