Stochastic differential equations driven by $G$-Brownian motion with reflecting boundary conditions

TL;DR

This paper develops stochastic calculus tools for $G$-Brownian motion with reflecting boundaries and investigates the solvability of related stochastic differential equations within the $G$-framework.

Contribution

It introduces stochastic integrals with respect to increasing processes in the $G$-framework and extends $G$-Itô's formula, addressing RGSDEs with reflecting boundaries.

Findings

Extended $G$-Itô's formula for new stochastic integrals.

Proved existence and uniqueness of solutions to RGSDEs.

Established foundational tools for $G$-stochastic calculus with boundaries.

Abstract

In this paper, we introduce the idea of stochastic integrals with respect to an increasing process in the $G$-framework and extend $G$-It\^o's formula. Moreover, we study the solvability of the scalar valued stochastic differential equations driven by $G$-Brownian motion with reflecting boundary conditions (RGSDEs).