Invariance principles for G-brownian-motion-driven stochastic differential equations and their applications to G-stochastic control

TL;DR

This paper develops new theoretical tools to analyze the long-term behavior of G-Brownian-motion-driven stochastic differential equations and applies these results to G-stochastic control problems.

Contribution

It introduces a new G-semimartingale convergence theorem and invariance principle, advancing the understanding of G-SDEs' long-time behaviors and their control applications.

Findings

Established a new G-semimartingale convergence theorem

Proved a new invariance principle for G-SDEs

Validated existence and uniqueness of solutions under local Lipschitz conditions

Abstract

The G-Brownian-motion-driven stochastic differential equations (G-SDEs) as well as the G-expectation, which were seminally proposed by Peng and his colleagues, have been extensively applied to describing a particular kind of uncertainty arising in real-world systems modeling. Mathematically depicting long-time and limit behaviors of the solution produced by G-SDEs is beneficial to understanding the mechanisms of system's evolution. Here, we develop a new G-semimartingale convergence theorem and further establish a new invariance principle for investigating the long-time behaviors emergent in G-SDEs. We also validate the uniqueness and the global existence of the solution of G-SDEs whose vector fields are only locally Lipchitzian with a linear upper bound. To demonstrate the broad applicability of our analytically established results, we investigate its application to achieving G-stochastic control in a few representative dynamical systems.