Convergence of the split-step θ-method for stochastic age-dependent population equations with Markovian switching and variable delay

Deng, Sen; Fei, Weiyin; Liang, Yong; Mao, Xuerong

doi:10.1016/j.apnum.2018.12.014

Cited by 11 publications

(4 citation statements)

References 26 publications

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“…For instance, we refer to [36] for the convergence in probability of EM method for highly nonlinear neutral stochastic differential equations with time-variable delay, and to [47] for the strong convergence and almost sure exponential stability of the backward EM method for nonlinear hybrid SDEs with time-variable delay. Further, Deng et al [14] discussed the strong convergence rate for the split-step theta method applied to stochastic age-dependent population equations with Markovian switching and time-variable delay.…”

Section: Introductionmentioning

confidence: 99%

The Truncated EM Method for Jump-Diffusion SDDEs with Super-Linearly Growing Diffusion and Jump Coefficients

Deng¹,

Fei²,

Mao³

2024

JCM

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This work is concerned with the convergence and stability of the truncated Euler-Maruyama (EM) method for super-linear stochastic differential delay equations (SDDEs) with time-variable delay and Poisson jumps. By constructing appropriate truncated functions to control the super-linear growth of the original coefficients, we present two types of the truncated EM method for such jump-diffusion SDDEs with time-variable delay, which is proposed to be approximated by the value taken at the nearest grid points on the left of the delayed argument. The first type is proved to have a strong convergence order which is arbitrarily close to 1/2 in mean-square sense, under the Khasminskii-type, global monotonicity with U function and polynomial growth conditions. The second type is convergent in qth (q < 2) moment under the local Lipschitz plus generalized Khasminskii-type conditions. In addition, we show that the partially truncated EM method preserves the mean-square and H∞ stabilities of the true solutions. Lastly, we carry out some numerical experiments to support the theoretical results.

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Section: Introductionmentioning

confidence: 99%

The Truncated EM Method for Jump-Diffusion SDDEs with Super-Linearly Growing Diffusion and Jump Coefficients

Deng¹,

Fei²,

Mao³

2024

JCM

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“…In recent years, the hybrid system driven by a continuous-time Markov chain has received a great deal of attention due to its important applications in economics, control, biology, finance and so on (see, e.g., Deng et al (2019); Mao (2002); Mao and Yuan (2006); Wang et al (2020)). One of the important works in these applications is to discuss the stability of a hybrid system.…”

Section: Introductionmentioning

confidence: 99%

Stabilization and destabilization of hybrid systems by periodic stochastic controls based on Lévy noise

Fei

Liang

et al. 2023

IMA Journal of Mathematical Control and Information

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We focus in this paper on determining whether or not a periodic stochastic feedback control based on Lévy noise can stabilize or destabilize a given non-linear hybrid system. Using the Lyapunov functions and the periodic functions, we establish some sufficient conditions on the stability and instability for non-linear hybrid systems with Lévy noise. Moreover, we use some numerical examples and simulations to illustrate that an unstable (or stable) non-linear hybrid system can be stabilized (or destabilized) via periodic stochastic feedback control based on Lévy noise.

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“…For instance, we refer to [29] for the convergence in probability of EM method for highly nonlinear neutral stochastic differential equations with variable delay, and to [30] for the strong error analysis of EM method for SDEs with variable and distributed delays. Further, in [31], strong convergence rates are derived for the split-step theta method applied to stochastic age-dependent population equations with Markovian switching and variable delay.…”

Section: Introductionmentioning

confidence: 99%

The truncated EM method for stochastic differential delay equations with variable delay

Deng,

Fei,

Fei

et al. 2021

Preprint

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This paper mainly investigates the strong convergence and stability of the truncated Euler-Maruyama (EM) method for stochastic differential delay equations with variable delay whose coefficients can be growing super-linearly. By constructing appropriate truncated functions to control the super-linear growth of the original coefficients, we present a type of the truncated EM method for such SDDEs with variable delay, which is proposed to be approximated by the value taken at the nearest grid points on the left of the delayed argument. The strong convergence result (without order) of the method is established under the local Lipschitz plus generalized Khasminskii-type conditions and the optimal strong convergence order 1/2 can be obtained if the global monotonicity with U function and polynomial growth conditions are added to the assumptions. Moreover, the partially truncated EM method is proved to preserve the mean-square and H ∞ stabilities of the true solutions. Compared with the known results on the truncated EM method for SDDEs, a better order of strong convergence is obtained under more relaxing conditions on the coefficients, and more refined technical estimates are developed so as to overcome the challenges arising due to variable delay. Lastly, some numerical examples are utilized to confirm the effectiveness of the theoretical results.

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Convergence of the split-step θ-method for stochastic age-dependent population equations with Markovian switching and variable delay

Cited by 11 publications

References 26 publications

The Truncated EM Method for Jump-Diffusion SDDEs with Super-Linearly Growing Diffusion and Jump Coefficients

The Truncated EM Method for Jump-Diffusion SDDEs with Super-Linearly Growing Diffusion and Jump Coefficients

Stabilization and destabilization of hybrid systems by periodic stochastic controls based on Lévy noise

The truncated EM method for stochastic differential delay equations with variable delay

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