Stratonovich-type integral with respect to a general stochastic measure

TL;DR

This paper establishes the well-definedness of a symmetric Stratonovich-type integral with respect to a general stochastic measure and proves existence and uniqueness of solutions for related stochastic equations.

Contribution

It introduces a new integral definition for general stochastic measures and demonstrates its applicability to solving stochastic equations with existence and uniqueness results.

Findings

The symmetric integral is well defined for general stochastic measures.

Existence and uniqueness of solutions are proven for stochastic equations using this integral.

The integral extends classical Stratonovich calculus to broader stochastic measures.

Abstract

Let $\mu$ be a general stochastic measure, where we assume for $\mu$ only $\sigma$-additivity in probability and continuity of paths. We prove that the symmetric integral $\int_{[0,T]}f(\mu_t, t)\circ\,{\rm d}\mu_t$ is well defined. For stochastic equations with this integral, we obtain the existence and uniqueness of a solution.