Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler–Maruyama approximation

Abstract

a b s t r a c tThis paper may be considered as a natural sequel to the paper [M. Milošević, Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama method, Mathematical and Computer Modelling 54 (2011) 2235-2251]. In the present paper, global almost sure (a.s.) asymptotic exponential stability of the equilibrium solution for a class of neutral stochastic differential equations with timedependent delay is considered, under nonlinear growth conditions. Additionally, the moment estimates are established for solutions of equations of this type. Under more restrictive conditions, including the linear growth condition, we show that the appropriate Euler-Maruyama equilibrium solution is globally a.s. asymptotically exponentially stable. As expected, the whole consideration is affected by the presence and properties of the delay function. In that sense, the delayed terms are explicitly treated as arguments of the coefficients of the equation and, particularly, under the derivative of the state variable. Additionally, some requirements related to the rate of change of the delay function are imposed in order to provide the main results of the paper.

Almost sure exponential stability of the θ-Euler-Maruyama method for neutral stochastic differential equations with time-dependent delay when θ ∈ [0; 1 2]

“…On the basis of the papers [12] and [13], one can conclude that hypotheses A 2 − A 4 , together with the local Lipschitz condition on f and and the assumption (8), guarantee the existence and uniqueness of the global solution of Eq. 1, which is almost surely exponentially stable.…”

“…1, which is almost surely exponentially stable. Results from [13] suggested to employ A 4 , as well as the linear growth condition A 1 on the drift coefficient of Eq. 1, in order to prove the almost sure exponential stability of the θ-Euler-Maruyama solution.…”

Exponential stability of the exact solutions and θ-EM approximations to neutral SDDEs with Markov switching

“…[15,16,20,21,17]. However, as far as we know, there are few results on the exponential stability of the θ -EM approximations of NSDDEswMS except [22]. In [22], Milošević only presented sufficient conditions of the exponential stability property for the Euler-Maruyama equilibrium solution of the given NSDDEswMS.…”

“…Note that θ -EM includes the classical EM method (θ = 0), the backward EM method (θ = 1) and the so-called trapezoidal method (θ = 1 2 ). In [22], the author needed the linear growth condition on both f and g for the almost sure exponential stability for the special case of (1.5) and (1.6) (θ = 0).…”

Exponential stability of numerical solution to neutral stochastic functional differential equation

“…However, the study on stability of numerical method for neutral stochastic differential systems is relatively scarce due to their technical difficulties, which is the main topic of the present paper. Recently, Milošević [14] showed that almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay. To the best knowledge of authors, there is no work on the stability of numerical solution to NSFDE.…”