Exponential stability of implicit numerical solution for nonlinear neutral stochastic differential equations with time‐varying delay and poisson jumps

Abstract

The aim of this work is to investigate the exponential mean‐square stability for neutral stochastic differential equations with time‐varying delay and Poisson jumps. When all the drift, diffusion, and jumps coefficients are allowed to be nonlinear, the exponential mean‐square stability of the analytic solution to the equation is obtained. It is revealed that the implicit backward Euler–Maruyama numerical solution can reproduce the corresponding stability of the analytic solution under some given nonlinear conditions. It is different from the explicit Euler–Maruyama numerical solution whose stability depends on the linear growth condition. With some requirements related to the delayed function and the property of compensated Poisson process, we deal with time‐varying delay and Poisson jumps. One highly nonlinear example is given to confirm the effectiveness of our theory.

On stability of numerical solutions of neutral stochastic delay differential equations with time‐dependent delay

“…Note that the property 𝛿 ∶= sup t≥0 𝛿 ′ (t) < 1 is a key assumption in the preceding theorem and 𝛿 appears explicitly in condition (3.47). This assumption is also essential in [11]. Under several additional conditions, it is proved in [13,14] that the EM and BEM approximations can reproduce the exponential stability for sufficiently small step sizes.…”

“…Then the BEM approximation (2.14) is well-defined whenever 0 < 𝜀 < (1 − 𝜅)∕(2K 0 ). We refer to [11,Section 4] and [14, Lemma 1] for details. For the convergence of the EM and BEM approximations, we refer to [2,3,12,14].…”

“…We proceed to study the exponential stability of the exact solution and numerical approximations of Equation (4.1). Since sup t≥0 𝛿 ′ (t) = 1, Theorem 3.6 and the criteria in [11,13] do not applicable here.…”

“…In [4][5][6][7][8], the exponential stability for the Euler-Maruyama (EM), backward Euler-Maruyama (BEM), and theta Euler-Maruyama approximations have been investigated for SDDEs, neutral SDDEs, and SFDEs. In [9][10][11], numerical solutions together with the moment and almost sure exponential stability of numerical solutions of neutral SDDEs with jumps have been investigated. Numerical solutions and the almost sure exponential stability of numerical solutions of highly nonlinear SDDEs under Khasminskiii-type conditions have been studied in [12][13][14][15][16].…”

“…It should be emphasized that it is nontrivial to handle the time-dependent delay in neutral SDDEs. Particularly, in [11] and [12][13][14], in order to study exponential stability of exact and numerical solutions of neutral SDDEs with time-dependent delay 𝛿(t), an essential assumption is that 𝛿(t) is continuously differentiable and 𝛿 ∶= sup t 𝛿 ′ (t) < 1. Moreover, the value of 𝛿 plays a key role in their analysis and in the stability criteria.…”