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Convergence of solutions of mixed stochastic delay differential equations with applications

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  • Mishura, Yuliya
  • Shalaiko, Taras
  • Shevchenko, Georgiy

Abstract

The paper is concerned with a mixed stochastic delay differential equation involving both a Wiener process and a γ-Hölder continuous process with γ>1/2 (e.g. a fractional Brownian motion with Hurst parameter greater than 1/2). It is shown that its solution depends continuously on the coefficients and the initial data. Two applications of this result are given: the convergence of solutions to equations with vanishing delay to the solution of equation without delay and the convergence of Euler approximations for mixed stochastic differential equations. As a side result of independent interest, the integrability of solution to mixed stochastic delay differential equations is established.

Suggested Citation

  • Mishura, Yuliya & Shalaiko, Taras & Shevchenko, Georgiy, 2015. "Convergence of solutions of mixed stochastic delay differential equations with applications," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 487-497.
    • Handle: RePEc:eee:apmaco:v:257:y:2015:i:c:p:487-497
    DOI: 10.1016/j.amc.2015.01.019
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    References listed on IDEAS

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    1. Hu, Yaozhong & Nualart, David & Song, Xiaoming, 2008. "A singular stochastic differential equation driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2075-2085, October.
    2. Kubilius, K., 2002. "The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type," Stochastic Processes and their Applications, Elsevier, vol. 98(2), pages 289-315, April.
    3. Shevchenko, Georgiy & Shalaiko, Taras, 2013. "Malliavin regularity of solutions to mixed stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 83(12), pages 2638-2646.
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    Cited by:

    1. Falkowski, Adrian & Słomiński, Leszek, 2022. "SDEs with two reflecting barriers driven by semimartingales and processes with bounded p-variation," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 164-186.
    2. Falkowski, Adrian & Słomiński, Leszek, 2017. "SDEs with constraints driven by semimartingales and processes with bounded p-variation," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3536-3557.
    3. Caraballo, Tomás & Cortés, J.-C. & Navarro-Quiles, A., 2019. "Applying the random variable transformation method to solve a class of random linear differential equation with discrete delay," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 198-218.

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