Quasi-Sure Exponential Stability of Stochastic Differential Delay Systems Driven by G-Brownian Motion

Fei, Chen; Yang, Luzhen; Fei, Weiyin

doi:10.3390/sym17020214

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Open AccessArticle

Quasi-Sure Exponential Stability of Stochastic Differential Delay Systems Driven by G-Brownian Motion

by

Chen Fei

¹

,

Luzhen Yang

¹ and

Weiyin Fei

^2,*

¹

Business School, University of Shanghai for Science and Technology, Shanghai 200093, China

²

School of Mathematics-Physics and Finance, Anhui Polytechnic University, Wuhu 241000, China

^*

Author to whom correspondence should be addressed.

Symmetry 2025, 17(2), 214; https://doi.org/10.3390/sym17020214

Submission received: 16 December 2024 / Revised: 17 January 2025 / Accepted: 21 January 2025 / Published: 31 January 2025

(This article belongs to the Special Issue Symmetric or Asymmetric Distributions and Its Applications)

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Abstract

This paper focuses on the quasi-sure exponential stability of the stochastic differential delay equation driven by G-Brownian motion (SDDE-GBM):

d ξ (t) = f (t, ξ (t - κ_{1} (t))) d t + g (t, ξ (t - κ_{2} (t))) d Z (t),

where

κ_{1} (\cdot), κ_{2} (\cdot) : R_{+} \to [0, τ]

denote variable delays, and

Z (t)

denotes scalar G-Brownian motion, which has a symmetry distribution. It is shown that the SDDE-GBM is quasi-surely exponentially stable for each

τ > 0

bounded by

τ^{*}

, where the corresponding (non-delay) stochastic differential equation driven by G-Bronwian motion (SDE-GBM),

d η (t) = f (t, η (t)) d t + g (t, η (t)) d Z (t)

, is quasi-surely exponentially stable. Moreover, by solving the non-linear equation on

τ

, we can obtain the implicit lower bound

τ^{*}

. Finally, illustrating examples are provided.

Keywords: SDDE-GBM; quasi-sure exponential stability; G-Brownian motion; delay bound; Borel–Cantelli’s lemma

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MDPI and ACS Style

Fei, C.; Yang, L.; Fei, W. Quasi-Sure Exponential Stability of Stochastic Differential Delay Systems Driven by G-Brownian Motion. Symmetry 2025, 17, 214. https://doi.org/10.3390/sym17020214

AMA Style

Fei C, Yang L, Fei W. Quasi-Sure Exponential Stability of Stochastic Differential Delay Systems Driven by G-Brownian Motion. Symmetry. 2025; 17(2):214. https://doi.org/10.3390/sym17020214

Chicago/Turabian Style

Fei, Chen, Luzhen Yang, and Weiyin Fei. 2025. "Quasi-Sure Exponential Stability of Stochastic Differential Delay Systems Driven by G-Brownian Motion" Symmetry 17, no. 2: 214. https://doi.org/10.3390/sym17020214

APA Style

Fei, C., Yang, L., & Fei, W. (2025). Quasi-Sure Exponential Stability of Stochastic Differential Delay Systems Driven by G-Brownian Motion. Symmetry, 17(2), 214. https://doi.org/10.3390/sym17020214

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Quasi-Sure Exponential Stability of Stochastic Differential Delay Systems Driven by G-Brownian Motion

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