$ h $-stability for stochastic functional differential equation driven by time-changed Lévy process

Liping Xu; Zhi Li; Benchen Huang; Liping Xu; Zhi Li; Benchen Huang

doi:10.3934/math.20231168

AIMS Mathematics

2023, Volume 8, Issue 10: 22963-22983. doi: 10.3934/math.20231168

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Research article

$ h $-stability for stochastic functional differential equation driven by time-changed Lévy process

School of Information and Mathematics, Yangtze University, Jingzhou 434023, China

Received: 09 March 2023 Revised: 09 June 2023 Accepted: 23 June 2023 Published: 19 July 2023
MSC : 60G15, 60H05, 60H15

In this paper, we investigate a class of stochastic functional differential equations driven by the time-changed Lévy process. Using the Lyapunov technique, we obtain some sufficient conditions to ensure that the solutions of the considered equations are $ h $-stable in $ p $-th moment sense. Subsequently, using time-changed Itô formula and a proof by reduction ad absurdum, we capture some new criteria for the $ h $-stability in mean square of the considered equations. In the end, we analyze some illustrative examples to show the interest and usefulness of the major results.
- $ h $-stability,
- time-changed Lévy process,
- Lyapunov method,
- time-changed Itô formula
Citation: Liping Xu, Zhi Li, Benchen Huang. $ h $-stability for stochastic functional differential equation driven by time-changed Lévy process[J]. AIMS Mathematics, 2023, 8(10): 22963-22983. doi: 10.3934/math.20231168

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Abstract

In this paper, we investigate a class of stochastic functional differential equations driven by the time-changed Lévy process. Using the Lyapunov technique, we obtain some sufficient conditions to ensure that the solutions of the considered equations are $ h $-stable in $ p $-th moment sense. Subsequently, using time-changed Itô formula and a proof by reduction ad absurdum, we capture some new criteria for the $ h $-stability in mean square of the considered equations. In the end, we analyze some illustrative examples to show the interest and usefulness of the major results.

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