Stochastic functional differential equations driven by G-Browniain motion with monotone nonlinearity

TL;DR

This paper develops a theoretical framework for the existence and uniqueness of solutions to stochastic functional differential equations driven by G-Brownian motion, using Picard iteration and monotonicity assumptions.

Contribution

It introduces new existence and uniqueness results for G-Brownian driven equations under both strong and weak monotonicity conditions, including error estimates and exponential bounds.

Findings

Established existence and uniqueness of solutions.

Derived error estimates for Picard approximations.

Obtained exponential estimates under monotonicity conditions.

Abstract

By using the Picard iteration scheme, this article establishes the existence and uniqueness theory for solutions to stochastic functional differential equations driven by G-Browniain motion. Assuming the monotonicity conditions, the boundedness and existence-uniqueness results of solutions have been derived. The error estimation between Picard approximate solution $y^k(t)$ and exact solution $y(t)$ has been determined. The $L^2_G$ and exponential estimates have been obtained. The theory has been further generalized to weak monotonicity conditions. The existence, uniqueness and exponential estimate under the weak monotonicity conditions have been inaugurated.