PPubMed11 May 2018Cited 2×

Analysis of stability for stochastic delay integro-differential equations

Yu Zhang, Longsuo Li

Abstract

In this paper, we concern stability of numerical methods applied to stochastic delay integro-differential equations. For linear stochastic delay integro-differential equations, it is shown that the mean-square stability is derived by the split-step backward Euler method without any restriction on step-size, while the Euler-Maruyama method could reproduce the mean-square stability under a step-size constraint. We also confirm the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. The numerical experiments further verify the theoretical results.

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Split-step backward Euler is unconditionally mean‑square stable for stochastic delay integro‑differential equations, while Euler‑Maruyama needs a step‑size bound.

What’s new

i.Proved unconditional mean‑square stability of the split‑step backward Euler method for linear stochastic delay integro‑differential equations
ii.Derived explicit step‑size constraints (h₁, h₂) for mean‑square stability of the Euler‑Maruyama method
iii.Extended the stability analysis to nonlinear stochastic delay integro‑differential equations
iv.Provided numerical experiments illustrating the theoretical results
v.Presented formulas for admissible step sizes (h₀ = max{h₁, h₂})

How they did it

The work uses analytical derivations of mean‑square stability definitions, constructs Lyapunov‑type inequalities, and proves theorems for both linear and nonlinear cases. Stability conditions are expressed in terms of the coefficients α,β,γ,λ,μ,η and the delay τ. Numerical experiments with various step sizes (e.g., h=1/15, 1/5, 1/100, 1/10) validate the theory.

What they found

Split‑step backward Euler method is mean‑square stable for any step size h
Euler‑Maruyama method is mean‑square stable only if h ≤ h₀, where h₀ = max{h₁, h₂} with h₁ = -(2α+2|β|+2|γ|τ+|λ|+|μ|+|η|τ²)/( |α|+|β|+|γ|τ² ) and h₂ = min{ -1/α , -(2α+2|β|+2|γ|τ+|λ|+|μ|+|η|τ²)/(α+|β|+|γ|τ²) }
Numerical experiments show SSBE remains stable for h=1/15 and h=1/5, while EM becomes unstable for h larger than the derived bound
For the nonlinear example with parameters a₁=80, a₂=10, a₃=10, b₁=2, b₂=2, b₃=20, τ=1, the split‑step method is unstable when h=0.1 > h₀

Caveats

✕Analysis is limited to mean‑square stability; other stability notions are not addressed
✕Results rely on specific coefficient assumptions and condition (2), limiting applicability to more general equations
✕Step‑size bound for Euler‑Maruyama may be conservative and requires full knowledge of coefficients
✕Numerical validation is confined to a few illustrative examples

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