Exponential stability of θ-EM method for nonlinear stochastic Volterra integro-differential equations
- Resource Type
- Authors
- Mei Zhao; Siyuan Qi; Guangqiang Lan
- Source
- Applied Numerical Mathematics. 172:279-291
- Subject
- Mean square
Computational Mathematics
Numerical Analysis
Nonlinear system
Lemma (mathematics)
Exponential stability
Differential equation
Chebyshev's inequality
Applied Mathematics
Applied mathematics
Almost surely
Linear growth
Mathematics
- Language
- ISSN
- 0168-9274
Mean square exponential stability of both exact solutions and the corresponding θ-EM method for stochastic Volterra integro-differential equations are investigated in this paper. For 1 2 θ ≤ 1 , we prove that both exact solutions and the corresponding θ-EM method for stochastic Volterra integro-differential equations are mean square exponentially stable under the Khasminskii-type conditions. If 0 ≤ θ ≤ 1 2 , θ-EM method is mean square exponentially stable under the Khasminskii-type condition plus linear growth condition on f. By using Chebyshev inequality and Borel-Cantelli lemma, we can also prove that θ-EM method is almost surely exponentially stable. An example is provided to support our conclusions.