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Journal articles on the topic 'Stochastic Delay Differential Equations'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Yang, Fang, Chen Fang, and Xu Sun. "Marcus Stochastic Differential Equations: Representation of Probability Density." Mathematics 12, no. 19 (2024): 2976. http://dx.doi.org/10.3390/math12192976.

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Marcus stochastic delay differential equations are often used to model stochastic dynamical systems with memory in science and engineering. It is challenging to study the existence, uniqueness, and probability density of Marcus stochastic delay differential equations, due to the fact that the delays cause very complicated correction terms. In this paper, we identify Marcus stochastic delay differential equations with some Marcus stochastic differential equations without delays but subject to extra constraints. This helps us to obtain the following two main results: (i) we establish a sufficien

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2

Tunc, Cemil, and Zozan Oktan. "STABILITY AND BOUNDEDNESS OF STOCHASTIC INTEGRO-DELAY DIFFERENTIAL EQUATIONS." Journal of Mathematical Analysis 15, no. 5 (2024): 69–83. https://doi.org/10.54379/jma-2024-5-5.

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This work addresses stochastic integro-delay differential equations (SIDDEs) of second order with two constant delays. In the study, two new results including sufficient conditions on stochastic asymptotic stability and stochastic boundedness in probability of solutions of the given SIDDEs are proved. The proofs of new results are done by using a Lyapunov-Krasovskii functional (L-KF) as a basic tool. To demonstrate the validity of the obtained results, two examples are provided. According to a comparison with previous literature, the results of this study are new and also allow new contributio

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3

Liu, Yue, Xuejing Meng, and Fuke Wu. "General Decay Stability for Stochastic Functional Differential Equations with Infinite Delay." International Journal of Stochastic Analysis 2010 (February 9, 2010): 1–17. http://dx.doi.org/10.1155/2010/875908.

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So far there are not many results on the stability for stochastic functional differential equations with infinite delay. The main aim of this paper is to establish some new criteria on the stability with general decay rate for stochastic functional differential equations with infinite delay. To illustrate the applications of our theories clearly, this paper also examines a scalar infinite delay stochastic functional differential equations with polynomial coefficients.

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4

Ma, Heping, Hui Jian, and Yu Shi. "A sufficient maximum principle for backward stochastic systems with mixed delays." Mathematical Biosciences and Engineering 20, no. 12 (2023): 21211–28. http://dx.doi.org/10.3934/mbe.2023938.

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<abstract><p>In this paper, we study the problem of optimal control of backward stochastic differential equations with three delays (discrete delay, moving-average delay and noisy memory). We establish the sufficient optimality condition for the stochastic system. We introduce two kinds of time-advanced stochastic differential equations as the adjoint equations, which involve the partial derivatives of the function $ f $ and its Malliavin derivatives. We also show that these two kinds of adjoint equations are equivalent. Finally, as applications, we discuss a linear-quadratic backw

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5

Petryna, G., and A. Stanzhytskyi. "APPROXIMATION OF STOCHASTIC DELAY DIFFERENTIAL SYSTEMS BY A STOCHASTIC SYSTEM WITHOUT DELAY." Bukovinian Mathematical Journal 12, no. 1 (2024): 120–36. http://dx.doi.org/10.31861/bmj2024.01.11.

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In this paper, we propose a scheme for approximating the solutions of stochastic differential equations with delay by solutions of stochastic differential equations without delay. Stochastic delay differential equations play a crucial role in modeling real-world processes where the evolution depends on past states, introducing complexities due to their infinite-dimensional phase space. To overcome these difficulties, we develop an approach based on approximating the delay system by an ordinary differential equation system of increased dimension. Our main result is to prove that, under certain

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6

Shevchenko, G. "Mixed stochastic delay differential equations." Theory of Probability and Mathematical Statistics 89 (January 26, 2015): 181–95. http://dx.doi.org/10.1090/s0094-9000-2015-00944-3.

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7

Pramila, G., and S. Ramadevi. "Time Delay and Mean Square Stochastic Differential Equations in Impetuous Stabilization." International Journal of Trend in Scientific Research and Development Volume-2, Issue-3 (2018): 627–31. http://dx.doi.org/10.31142/ijtsrd11062.

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8

Fofana, M. S. "Moment Lyapunov exponent of delay differential equations." International Journal of Mathematics and Mathematical Sciences 30, no. 6 (2002): 339–51. http://dx.doi.org/10.1155/s0161171202012103.

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The aim of this paper is to establish a connecting thread through the probabilistic concepts ofpth-moment Lyapunov exponents, the integral averaging method, and Hale's reduction approach for delay dynamical systems. We demonstrate this connection by studying the stability of perturbed deterministic and stochastic differential equations with fixed time delays in the displacement and derivative functions. Conditions guaranteeing stable and unstable solution response are derived. It is felt that the connecting thread provides a unified framework for the stability study of delay differential equat

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9

Wang, Peiguang, and Yan Xu. "Averaging Method for Neutral Stochastic Delay Differential Equations Driven by Fractional Brownian Motion." Journal of Function Spaces 2020 (May 29, 2020): 1–7. http://dx.doi.org/10.1155/2020/5212690.

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In this paper, we investigate the stochastic averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H∈1/2,1. By using the linear operator theory and the pathwise approach, we show that the solutions of neutral stochastic delay differential equations converge to the solutions of the corresponding averaged stochastic delay differential equations. At last, an example is provided to illustrate the applications of the proposed results.

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10

Ponosov, Arcady V. "Existence and uniqueness of solutions to stochastic fractional differential equations in multiple time scales." Russian Universities Reports. Mathematics, no. 141 (2023): 51–59. http://dx.doi.org/10.20310/2686-9667-2023-28-141-51-59.

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A novel class of nonlinear stochastic fractional differential equations with delay and the Jumarie and Ito differentials is introduced in the paper. The aim of the study is to prove existence and uniqueness of solutions to these equations. The main results of the paper generalise some previous findings made for the non-delay and three-scale equations under additional restrictions on the fractional order of the Jumarie differentials, which are removed in our analysis. The techniques used in the paper are based on the properties of the singular integral operators in specially designed spaces of

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11

Catuogno, Pedro, and Paulo Ruffino. "Geometry of Stochastic Delay Differential Equations." Electronic Communications in Probability 10 (2005): 190–95. http://dx.doi.org/10.1214/ecp.v10-1151.

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12

Chen, Li, Zhen Wu, and Zhiyong Yu. "Delayed Stochastic Linear-Quadratic Control Problem and Related Applications." Journal of Applied Mathematics 2012 (2012): 1–22. http://dx.doi.org/10.1155/2012/835319.

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We discuss a quadratic criterion optimal control problem for stochastic linear system with delay in both state and control variables. This problem will lead to a kind of generalized forward-backward stochastic differential equations (FBSDEs) with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the optimal feedback regulator for the time delay system via a new type of Riccati equations and also apply to a population optimal control problem.

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13

Guillouzic, Steve, Ivan L’Heureux, and André Longtin. "Small delay approximation of stochastic delay differential equations." Physical Review E 59, no. 4 (1999): 3970–82. http://dx.doi.org/10.1103/physreve.59.3970.

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14

Buckwar, Evelyn, Rachel Kuske, Salah-Eldin Mohammed, and Tony Shardlow. "Weak Convergence of the Euler Scheme for Stochastic Differential Delay Equations." LMS Journal of Computation and Mathematics 11 (2008): 60–99. http://dx.doi.org/10.1112/s146115700000053x.

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AbstractWe study weak convergence of an Euler scheme for nonlinear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is non-anticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

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15

G., Pramila, and Ramadevi S. "Time Delay and Mean Square Stochastic Differential Equations in Impetuous Stabilization." International Journal of Trend in Scientific Research and Development 2, no. 3 (2018): 627–31. https://doi.org/10.31142/ijtsrd11062.

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This paper specially exhibits about the time delay and mean square stochastic differential equations in impetuous stabilization is analyzed. By projecting a delay differential inequality and using the stochastic analysis technique, a few present stage for mean square exponential stabilization are survived. It is express that an unstable stochastic delay system can be achieved some stability by impetuous. This example is also argued to derived the efficiency of the obtained results. G. Pramila | S. Ramadevi "Time Delay and Mean Square Stochastic Differential Equations in Impetuous Stabiliz

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16

Wang, Peiguang, and Yan Xu. "Periodic Averaging Principle for Neutral Stochastic Delay Differential Equations with Impulses." Complexity 2020 (June 15, 2020): 1–10. http://dx.doi.org/10.1155/2020/6731091.

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In this paper, we study the periodic averaging principle for neutral stochastic delay differential equations with impulses under non-Lipschitz condition. By using the linear operator theory, we deal with the difficulty brought by delay term of the neutral system and obtain the conclusion that the solutions of neutral stochastic delay differential equations with impulses converge to the solutions of the corresponding averaged stochastic delay differential equations without impulses in the sense of mean square and in probability. At last, an example is presented to show the validity of the propo

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17

Sarhan, Falah, and LIU JICHENG. "Euler-Maruyama approximation of backward doubly stochastic differential delay equations." International Journal of Applied Mathematical Research 5, no. 3 (2016): 146. http://dx.doi.org/10.14419/ijamr.v5i3.6358.

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In this paper, we attempt to introduce a new numerical approach to solve backward doubly stochastic differential delay equation ( shortly-BDSDDEs ). In the beginning, we present some assumptions to get the numerical scheme for BDSDDEs, from which we prove important theorem. We use the relationship between backward doubly stochastic differential delay equations and stochastic controls by interpreting BDSDDEs as some stochastic optimal control problems, to solve the approximated BDSDDEs and we prove that the numerical solutions of backward doubly stochastic differential delay equation converge t

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18

Tagelsir, A. Ahmed*, and Casteren Jan A. Van. "DENSITIES OF DISTRIBUTIONS OF SOLUTIONS TO DELAY STOCHASTIC DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS INITIAL DATA ( PART II)." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 5, no. 3 (2016): 530–42. https://doi.org/10.5281/zenodo.47697.

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In the present work we have gone a step forward towards integration by part of higher order Malliavin derivatives by formulating and extending some formulas and results on Malliavin calculus and ordinary stochastic differential equations to include delay stochastic differential equations as well as ordinary SDE’s. Here we have also stated clearly what we mean by the Malliavin derivatives and densities of distributions of the solutions process for delay stochastic differential equations which we are considering.

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19

Alsaadi, Fawaz E., Lichao Feng, Madini O. Alassafi, Reem M. Alotaibi, Adil M. Ahmad, and Jinde Cao. "Stochastic Robustness of Delayed Discrete Noises for Delay Differential Equations." Mathematics 10, no. 5 (2022): 743. http://dx.doi.org/10.3390/math10050743.

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Stochastic robustness of discrete noises has already been proposed and studied in the previous work. Nevertheless, the significant phenomenon of delays is left in the basket both in the deterministic and the stochastic parts of the considered equation by the existing work. Stimulated by the above, this paper is devoted to studying the stochastic robustness issue of delayed discrete noises for delay differential equations, including the issues of robust stability and robust boundedness.

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20

Sykora, Henrik T., Daniel Bachrathy, and Gabor Stepan. "Stochastic semi‐discretization for linear stochastic delay differential equations." International Journal for Numerical Methods in Engineering 119, no. 9 (2019): 879–98. http://dx.doi.org/10.1002/nme.6076.

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21

Mao, Xuerong, Alexander Matasov, and Aleksey B. Piunovskiy. "Stochastic Differential Delay Equations with Markovian Switching." Bernoulli 6, no. 1 (2000): 73. http://dx.doi.org/10.2307/3318634.

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22

KUTOYANTS, YURY A. "ON DELAY ESTIMATION FOR STOCHASTIC DIFFERENTIAL EQUATIONS." Stochastics and Dynamics 05, no. 02 (2005): 333–42. http://dx.doi.org/10.1142/s0219493705001444.

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We present a review of some results concerning delay estimation by continuous time observations of solutions of stochastic differential equations in two asymptotics. The first one corresponds to small noise limit and the second to large samples limit. In both cases we describe the properties of the maximum likelihood estimator and Bayesian estimators with especial attention to asymptotic efficiency of the estimators. We show that the first asymptotic corresponds to regular problems of mathematical statistics and the second is close to non regular problems. In small noise asymptotics we give th

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23

Mao, Xuerong, and Anita Shah. "Exponential stability of stochastic differential delay equations." Stochastics and Stochastic Reports 60, no. 1-2 (1997): 135–53. http://dx.doi.org/10.1080/17442509708834102.

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24

Krapivsky, P. L., J. M. Luck, and K. Mallick. "On stochastic differential equations with random delay." Journal of Statistical Mechanics: Theory and Experiment 2011, no. 10 (2011): P10008. http://dx.doi.org/10.1088/1742-5468/2011/10/p10008.

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25

Mo, Chi, and Jiaowan Luo. "Large deviations for stochastic differential delay equations." Nonlinear Analysis: Theory, Methods & Applications 80 (March 2013): 202–10. http://dx.doi.org/10.1016/j.na.2012.10.004.

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26

El-Borai, Mahmoud M., Khairia El-Said El-Nadi, and Hoda A. Fouad. "On some fractional stochastic delay differential equations." Computers & Mathematics with Applications 59, no. 3 (2010): 1165–70. http://dx.doi.org/10.1016/j.camwa.2009.05.004.

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27

Zhou, Shaobo, Zhiyong Wang, and Dan Feng. "Stochastic functional differential equations with infinite delay." Journal of Mathematical Analysis and Applications 357, no. 2 (2009): 416–26. http://dx.doi.org/10.1016/j.jmaa.2009.04.015.

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28

Buckwar, E. "Weak approximation of stochastic differential delay equations." IMA Journal of Numerical Analysis 25, no. 1 (2005): 57–86. http://dx.doi.org/10.1093/imanum/drh012.

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29

Shen, Yi, and Xuerong Mao. "ASYMPTOTIC BEHAVIOURS OF STOCHASTIC DIFFERENTIAL DELAY EQUATIONS." Asian Journal of Control 8, no. 1 (2008): 21–27. http://dx.doi.org/10.1111/j.1934-6093.2006.tb00247.x.

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30

Klosek, M. M., and R. Kuske. "Multiscale Analysis of Stochastic Delay Differential Equations." Multiscale Modeling & Simulation 3, no. 3 (2005): 706–29. http://dx.doi.org/10.1137/030601375.

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31

Mao, Xuerong, Chenggui Yuan, and Jiezhong Zou. "Stochastic differential delay equations of population dynamics." Journal of Mathematical Analysis and Applications 304, no. 1 (2005): 296–320. http://dx.doi.org/10.1016/j.jmaa.2004.09.027.

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32

Zhu, Qingfeng, Yufeng Shi, Jiaqiang Wen, and Hui Zhang. "A Type of Time-Symmetric Stochastic System and Related Games." Symmetry 13, no. 1 (2021): 118. http://dx.doi.org/10.3390/sym13010118.

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This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differ

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33

Øksendal, Bernt, Agnès Sulem, and Tusheng Zhang. "Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations." Advances in Applied Probability 43, no. 2 (2011): 572–96. http://dx.doi.org/10.1239/aap/1308662493.

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We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

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34

Øksendal, Bernt, Agnès Sulem, and Tusheng Zhang. "Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations." Advances in Applied Probability 43, no. 02 (2011): 572–96. http://dx.doi.org/10.1017/s0001867800004997.

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We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

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35

Hu, Peng, та Chengming Huang. "Stability of stochasticθ-methods for stochastic delay integro-differential equations". International Journal of Computer Mathematics 88, № 7 (2011): 1417–29. http://dx.doi.org/10.1080/00207160.2010.509430.

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36

S., Elizabeth, and Nirmal Veena S. "STABILIZATION OF DISCRETE STOCHASTIC DYNAMIC SYSTEM WITH DELAY." International Journal of Current Research and Modern Education, Special Issue (August 13, 2017): 53–56. https://doi.org/10.5281/zenodo.842234.

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In this paper, we discuss the stability of stochastic type differential equations through obtaining the stability condition for the respective stochastic difference equation. The system formulation is done by considering the stochastic differential equation that describes the dynamics of single isolated neuron involving delay. Here the discretization of the stochastic differential equation is done through the Euler- Maruyama Method. And the desired stability is obtained by applying suitable assumptions and through the help of theorems. The obtained theoretical results are represented through n

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37

Zhang, Wei, M. H. Song, and M. Z. Liu. "Almost sure exponential stability of stochastic differential delay equations." Filomat 33, no. 3 (2019): 789–814. http://dx.doi.org/10.2298/fil1903789z.

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This paper mainly studies whether the almost sure exponential stability of stochastic differential delay equations (SDDEs) is shared with that of the stochastic theta method. We show that under the global Lipschitz condition the SDDE is pth moment exponentially stable (for p 2 (0; 1)) if and only if the stochastic theta method of the SDDE is pth moment exponentially stable and pth moment exponential stability of the SDDE or the stochastic theta method implies the almost sure exponential stability of the SDDE or the stochastic theta method, respectively. We then replace the global Lipschitz con

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38

KUSKE, R. "MULTI-SCALE DYNAMICS IN STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE." Stochastics and Dynamics 05, no. 02 (2005): 233–46. http://dx.doi.org/10.1142/s0219493705001390.

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We apply multi-scale analysis to stochastic delay-differential equations with multiplicative or parametric noise, deriving approximate stochastic equations for the amplitudes of oscillatory solutions near critical delays. Reduced equations for the envelope of the oscillations provides an efficient analysis of the dynamics by separating the influence of the noise from the intrinsic oscillations over long time scales. We show how this analysis can be used to compute Lyapunov exponents and extended to nonlinear models where the noise has additional resonances.

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39

Hu, Rong, and De Jun Shao. "Stability of Neutral Stochastic Delay Differential Equations with Infinite Delay." Applied Mechanics and Materials 288 (February 2013): 105–8. http://dx.doi.org/10.4028/www.scientific.net/amm.288.105.

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This paper considers the pth moment stability of solution to neutral stochastic delay differential equation with infinite delay with local Lipschitz condition but neither the linear growth condition. The stability is more general and representative than the exponential stability. This investigation uses a specific Lyapunov function based on usual methods. An example is discussed to illustrate the theory.

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40

Balasubramaniam, P., and J. P. Dauer. "Controllability of semilinear stochastic delay evolution equations in Hilbert spaces." International Journal of Mathematics and Mathematical Sciences 31, no. 3 (2002): 157–66. http://dx.doi.org/10.1155/s0161171202111318.

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The controllability of semilinear stochastic delay evolution equations is studied by using a stochastic version of the well-known Banach fixed point theorem and semigroup theory. An application to stochastic partial differential equations is given.

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41

Li, Yan, and Junhao Hu. "Numerical Analysis for Stochastic Partial Differential Delay Equations with Jumps." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/128625.

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We investigate the convergence rate of Euler-Maruyama method for a class of stochastic partial differential delay equations driven by both Brownian motion and Poisson point processes. We discretize in space by a Galerkin method and in time by using a stochastic exponential integrator. We generalize some results of Bao et al. (2011) and Jacob et al. (2009) in finite dimensions to a class of stochastic partial differential delay equations with jumps in infinite dimensions.

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42

REISS, MARKUS. "ESTIMATING THE DELAY TIME IN AFFINE STOCHASTIC DELAY DIFFERENTIAL EQUATIONS." International Journal of Wavelets, Multiresolution and Information Processing 02, no. 04 (2004): 525–44. http://dx.doi.org/10.1142/s0219691304000664.

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We consider linear differential equations with bounded time delay driven by additive white noise. Our aim is the estimation of the maximal delay time from observations of one realisation of the solution process X under nonparametric drift assumptions. In the stationarity case the covariance function has a jump in the third derivative according to the location of the delay time. Based on this result, the delay time estimator is obtained from a singularity detection in the covariance function using a multiresolution framework. It is proved that the estimator attains the rate T-1/3 for observatio

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43

Diomande, Bakarime, and Lucian Maticiuc. "Multivalued stochastic delay differential equations and related stochastic control problems." Quaestiones Mathematicae 40, no. 6 (2017): 769–802. http://dx.doi.org/10.2989/16073606.2017.1315346.

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44

Babaei, Afshin, Sedigheh Banihashemi, Behrouz Parsa Moghaddam, Arman Dabiri, and Alexandra Galhano. "Efficient Solutions for Stochastic Fractional Differential Equations with a Neutral Delay Using Jacobi Poly-Fractonomials." Mathematics 12, no. 20 (2024): 3273. http://dx.doi.org/10.3390/math12203273.

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This paper introduces a novel numerical technique for solving fractional stochastic differential equations with neutral delays. The method employs a stepwise collocation scheme with Jacobi poly-fractonomials to consider unknown stochastic processes. For this purpose, the delay differential equations are transformed into augmented ones without delays. This transformation makes it possible to use a collocation scheme improved with Jacobi poly-fractonomials to solve the changed equations repeatedly. At each iteration, a system of nonlinear equations is generated. Next, the convergence properties

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45

Luo, Chaoliang, Shangjiang Guo, and Aiyu Hou. "Bifurcation of a class of stochastic delay differential equations." Filomat 34, no. 6 (2020): 1821–34. http://dx.doi.org/10.2298/fil2006821l.

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In this paper, we study the bifurcation of a class of two-dimensional stochastic delay differential equations. Firstly, we translate the original system into an It? limiting diffusion system by applying stochastic Taylor expansion, small time delay expansion, polar coordinate transformation, and stochastic averaging procedure. Then we discuss the dynamical bifurcation by analyzing the qualitative changes of invariant measures, and investigate the phenomenological bifurcation by utilizing Fokker-Planck equation. The obtained conclusions are completely new, which generalize and improve some exis

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46

SRI NAMACHCHIVAYA, N., and VOLKER WIHSTUTZ. "ALMOST SURE ASYMPTOTIC STABILITY OF SCALAR STOCHASTIC DELAY EQUATIONS: FINITE STATE MARKOV PROCESS." Stochastics and Dynamics 12, no. 01 (2012): 1150010. http://dx.doi.org/10.1142/s0219493712003560.

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In this paper, we study the almost-sure asymptotic stability of scalar delay differential equations with random parametric fluctuations which are modeled by a Markov process with finitely many states. The techniques developed for the determination of almost-sure asymptotic stability of finite dimensional stochastic differential equations will be extended to delay differential equations with random parametric fluctuations. For small intensity noise, we construct an asymptotic expansion for the exponential growth rate (the maximal Lyapunov exponent), which determines the almost-sure stability of

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47

Tergeussizova, Aliya. "MATHEMATICAL MODELING OF THE PROCESS OF DRAWING AN OPTICAL FIBER USING THE LANGEVIN EQUATION." Informatyka Automatyka Pomiary w Gospodarce i Ochronie Środowiska 9, no. 2 (2019): 64–67. http://dx.doi.org/10.5604/01.3001.0013.2551.

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In order to design stochastic pulse frequency systems for automatic control of objects with delay, this article shows how we obtained their models in the form of stochastic differential equations. The method of dynamic compensation of objects with delay is considered. A stochastic differential system in the Langevin form is obtained.

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48

Liu, Guodong, Kaiyuan Liu, and Xinzhu Meng. "Dynamical Analysis and Optimal Harvesting Strategy for a Stochastic Delayed Predator-Prey Competitive System with Lévy Jumps." Mathematical Problems in Engineering 2019 (February 12, 2019): 1–14. http://dx.doi.org/10.1155/2019/2187274.

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This paper develops a theoretical framework to investigate optimal harvesting control for stochastic delay differential systems. We first propose a novel stochastic two-predator and one-prey competitive system subject to time delays and Lévy jumps. Then we obtain sufficient conditions for persistence in mean and extinction of three species by using the stochastic qualitative analysis method. Finally, the optimal harvesting effort and the maximum of expectation of sustainable yield (ESY) are derived from Hessian matrix method and optimal harvesting theory of delay differential equations. Moreov

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49

Cao, Wanrong, and Zhongqiang Zhang. "Simulations of Two-Step Maruyama Methods for Nonlinear Stochastic Delay Differential Equations." Advances in Applied Mathematics and Mechanics 4, no. 06 (2012): 821–32. http://dx.doi.org/10.4208/aamm.12-12s11.

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AbstractIn this paper, we investigate the numerical performance of a family ofP-stable two-step Maruyama schemes in mean-square sense for stochastic differential equations with time delay proposed in for a certain class of nonlinear stochastic delay differential equations with multiplicative white noises. We also test the convergence of one of the schemes for a time-delayed Burgers’ equation with an additive white noise. Numerical results show that this family of two-step Maruyama methods exhibit similar stability for nonlinear equations as that for linear equations.

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BAKER, CHRISTOPHER T. H., JUDITH M. FORD, and NEVILLE J. FORD. "BIFURCATIONS IN APPROXIMATE SOLUTIONS OF STOCHASTIC DELAY DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 14, no. 09 (2004): 2999–3021. http://dx.doi.org/10.1142/s0218127404011235.

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Abstract:

We consider stochastic delay differential equations of the form [Formula: see text] interpreted in the Itô sense, with Y(t)=Φ(t) for t∈[t0-τ,t0] (here, W(t) is a standard Wiener process and τ>0 is the constant "lag", or "time-lag"). We are interested in bifurcations (that is, changes in the qualitative behavior of solutions of these equations) and we draw on insights from the related deterministic delay differential equation, for which there is a substantial body of known theory, and numerical results that enable us to discuss where changes occur in the behavior of the (exact and approximat

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