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

Xiaoxiao Peng, Shijie Zhou, Wei Lin, Xuerong Mao

TL;DR

The work develops a framework for analyzing long-time behavior of stochastic systems under uncertainty by introducing a $G$-semimartingale convergence theorem and an invariance principle within the $G$-expectation setting. It establishes global well-posedness for $G$-SDEs with locally Lipschitz, linear-growth vector fields and demonstrates how these invariance principles enable robust $G$-stochastic control in representative dynamical systems. The results bridge classical stochastic stability concepts with sublinear expectation theory, providing quasi-sure convergence guarantees and Lyapunov-based stability under uncertain noise. The findings have practical implications for controlling and understanding complex systems subject to model uncertainty, with potential extensions to $G$-SDDEs/$G$-SFDEs and rigorous numerical schemes.

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.

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

TL;DR

The work develops a framework for analyzing long-time behavior of stochastic systems under uncertainty by introducing a -semimartingale convergence theorem and an invariance principle within the -expectation setting. It establishes global well-posedness for -SDEs with locally Lipschitz, linear-growth vector fields and demonstrates how these invariance principles enable robust -stochastic control in representative dynamical systems. The results bridge classical stochastic stability concepts with sublinear expectation theory, providing quasi-sure convergence guarantees and Lyapunov-based stability under uncertain noise. The findings have practical implications for controlling and understanding complex systems subject to model uncertainty, with potential extensions to -SDDEs/-SFDEs and rigorous numerical schemes.

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.
Paper Structure (14 sections, 28 theorems, 32 equations, 1 figure)

This paper contains 14 sections, 28 theorems, 32 equations, 1 figure.

Table of Contents

  1. abstract
  2. Introduction
  3. Preliminaries
  4. $G$-Semimartingale Convergence Theorem
  5. Invariance Principle in Sublinear Expectation
  6. Illustrative Examples: Applying $G$-invariance principle to achieving $G$-stochastic control
  7. Conclusion
  8. Appendix
  9. Proof of Proposition \ref{['convergence']}
  10. Proof of Proposition \ref{['exist2']}
  11. Proof of Lemma \ref{['theorembarbalat']}
  12. Dynamic Stability in Example \ref{['exstability']}
  13. Invariant Set Associated with Autonomous $G$-SDEs
  14. Numerical evidences

Key Result

THEOREM 2.1

Assume $A^{1}$ and $A^{2}$ are two non-decreasing process with initial value 0, $A^{1}(t)$ is a continuous process and $\mathbb{\hat{E}}[A^{1}(+\infty)]<+\infty$. Assume that $Z$ is a non-negative $G$-semimartingale satisfying $\mathbb{\hat{E}}[Z^{+}({0})]<\infty$ with the form as $Z({t})=Z({0})+A^{

Figures (1)

  • Figure 1: (a)The dynamics of ${\rm log}\vert {\bm x}\vert$ change with $t$ for a group of SDEs correspondingly from the $G$-SDEs in Example \ref{['example1']}. Here, simulated are the 400 trials using the settings, $\underline{\sigma}^2=3.5$, $\overline{\sigma}^2=4$. (b)The dynamics of $\vert\bm x\vert$ change with $t$ for a group of SDEs correspondingly from the $G$-SDEs in Example \ref{['exstability']}. Here, simulated are the 400 trials using the settings: $\underline{\sigma}^2=40$, $\overline{\sigma}^2=50$, $\sigma=10$, $\rho=10$, $\beta=8/3$, and $k=5$. (c)The dynamics of $\vert\bm x\vert$ change with $t$ for a group of SDEs correspondingly from the $G$-SDEs in Example \ref{['example3']}. Here, simulated are the 400 trials using the settings: $\underline{\sigma}^2=40$ and $\overline{\sigma}^2=50$.

Theorems & Definitions (41)

  • THEOREM 2.1
  • THEOREM 2.2
  • Definition 3.1: Sublinear Expectation Peng-13
  • Definition 3.2: $G$-Function Peng-13
  • THEOREM 3.3: HuPeng-29
  • Lemma 3.4: Peng-13
  • Remark 3.5
  • THEOREM 3.6
  • Definition 3.7: Choquet Capacity, Peng-13
  • Proposition 3.8: Monotone Convergence Theorem, LaurentDenis-25Peng-26
  • ...and 31 more