On the strong Markov property for stochastic differential equations driven by $G$-Brownian motion

TL;DR

This paper extends the strong Markov property to stochastic differential equations driven by $G$-Brownian motion, broadening the understanding of their probabilistic structure and enabling new applications like the reflection principle.

Contribution

It develops the strong Markov property for $G$-SDEs by extending conditional $G$-expectation to optional times, including for $G$-Brownian motion.

Findings

Established the strong Markov property for $G$-Brownian motion.

Extended conditional $G$-expectation to optional times.

Provided applications such as the reflection principle.

Abstract

In this paper we study the stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs for short). We extend the notion of conditional $G$-expectation from deterministic time to the more general optional time situation. Then, via this conditional expectation, we develop the strong Markov property for $G$-SDEs. In particular, we obtain the strong Markov property for $G$-Brownian motion. Some applications including the reflection principle for $G$-Brownian motion are also provided.