Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler–Maruyama method

Milošević, Marija

doi:10.1016/j.mcm.2011.05.033

Cited by 45 publications

(35 citation statements)

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“…In [10], the existence and uniqueness of the solution is proved under these conditions, as well as the convergence in probability of the appropriate Euler-Maruyama solution. So, the main aim of this paper is to prove global a.s. asymptotic exponential stability of the exact equilibrium solution to an equation of this type and to determine the conditions under which the Euler-Maruyama equilibrium solution has the same property.…”

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confidence: 99%

“…This paper, which is natural extension of the paper [10], is organized in the following way. After introducing the basic notation and results which are explicitly used in the further analysis, in Section 2 we prove the global a.s. asymptotic exponential stability of the exact equilibrium solution to a class of neutral stochastic differential equations with time-dependent delay, under nonlinear growth conditions.…”

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confidence: 99%

“…After introducing the basic notation and results which are explicitly used in the further analysis, in Section 2 we prove the global a.s. asymptotic exponential stability of the exact equilibrium solution to a class of neutral stochastic differential equations with time-dependent delay, under nonlinear growth conditions. In Section 3, we introduce some more restrictive conditions which are sufficient for global a.s. asymptotic exponential stability of the appropriate Euler-Maruyama solution considered in [10].…”

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confidence: 99%

“…Since this paper represents a sequel of the paper [10], the notation and definitions will be mostly the same as in that cited paper. However, they will be introduced here for completeness of this paper.…”

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

Milošević

2013

Mathematical and Computer Modelling

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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.

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

Milošević

2013

Mathematical and Computer Modelling

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“…For example, Wu and Mao [11] studied the convergence of EM approximation for NSFDE with the linear growth coefficients. Milošević [12] studied convergence in probability of the EM approximate solutions to nonlinear neutral SDDEs under the Khasminskii-type conditions. Zhou and Fang [13] established convergence in probability of the EM approximate solutions for highly nonlinear NSFDEs.…”

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Exponential stability of numerical solution to neutral stochastic functional differential equation

Zhou

2015

Applied Mathematics and Computation

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

Liu

Xing

et al. 2020

Math Methods in App Sciences

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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.

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Highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler–Maruyama method

Cited by 45 publications

References 16 publications

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

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

Exponential stability of numerical solution to neutral stochastic functional differential equation

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

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