  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.
