Regularity of Solutions of Mean-Field $G$-SDEs

TL;DR

This paper investigates the regularity and differentiability properties of solutions to mean-field $G$-stochastic differential equations, establishing their first and second order Fréchet differentiability with respect to initial conditions.

Contribution

It introduces the first and second order Fréchet differentiability of solutions to mean-field $G$-SDEs and characterizes the associated $G$-SDEs for these derivatives.

Findings

Proves first and second order Fréchet differentiability of solutions.

Derives $G$-SDEs governing the derivatives.

Establishes regularity properties of mean-field $G$-SDE solutions.

Abstract

We study regularity properties of the unique solution of a mean-field $G$-SDE. More precisely, we consider a mean-field $G$-SDE with square-integrable random initial condition and establish its first and second order Fr\'echet differentiability in the random initial condition and specify the $G$-SDEs of the respective Fr\'echet derivatives.