On distribution dependent stochastic differential equations driven by $G$-Brownian motion
Abstract
Distribution dependent stochastic differential equations have been a very hot subject with extensive studies. On the other hand, under the -expectation framework, stochastic differential equations driven by -Brownian motion (in short form, -SDEs) have received increasing attentions, and the existence and uniqueness of solutions to -SDEs under Lipschitz and non-Lipschitz conditions have been obtained. Based on these studies, it is very natural and also important to investigate the -SDEs which are also distribution dependent. In this paper, we are concerned with the well-posedness of the distribution dependent -SDEs. To this end, we first introduce a proper distance of the involved distribution functions and propose a new formulation of the distribution dependent -SDEs. Then, by utilising fix point argument, we establish existence and uniqueness of the solutions of distributed dependent -SDEs under Lipschitz condition. Finally, we derive certain estimates for the solutions of the distribution dependent -SDEs.
Cite
@article{arxiv.2302.12539,
title = {On distribution dependent stochastic differential equations driven by $G$-Brownian motion},
author = {De Sun and Jiang-Lun Wu and Panyu Wu},
journal= {arXiv preprint arXiv:2302.12539},
year = {2023}
}