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

Peng, Xiaoxiao and Zhou, Shijie and Lin, Wei and Mao, Xuerong (2024) Invariance principles for G-Brownian-motion-driven stochastic differential equations and their applications to G-stochastic control. SIAM Journal on Control and Optimization. ISSN 0363-0129 (In Press)

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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 Lipschitzian 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.

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