Wong-Zakai Approximation for SDEs Driven by $G-$Brownian Motion

TL;DR

This paper establishes the equivalence between rough differential equations driven by lifted G-Brownian motion and Stratonovich SDEs, providing convergence rates for Wong-Zakai approximations under G-expectation framework.

Contribution

It introduces a Wong-Zakai approximation approach for G-SDEs and estimates the convergence rate in the G-framework, linking rough paths and stochastic calculus.

Findings

Wong-Zakai approximation converges quasi-surely with rate $(1/n)^{1/2-}$.

Established equivalence between rough differential equations and Stratonovich G-SDEs.

Proved quasi-sure continuity of solutions with respect to uniform norm.

Abstract

In this paper, we build the equivalence between rough differential equations driven by the lifted $G$-Brownian motion and the corresponding Stratonovich type SDE through the Wong-Zakai approximation. The quasi-surely convergence rate of Wong-Zakai approximation to $G-$SDEs with mesh-size $\frac{1}{n}$ in the $\alpha$-H\"older norm is estimated as $(\frac{1}{n})^{\frac12-}.$ As corollary, we obtain the quasi-surely continuity of the above RDEs with respect to uniform norm.