Mixed stochastic delay differential equations
Author:
G. Shevchenko
Journal:
Theor. Probability and Math. Statist. 89 (2014), 181-195
MSC (2010):
Primary 60H10, 34K50, 60G22
DOI:
https://doi.org/10.1090/S0094-9000-2015-00944-3
Published electronically:
January 26, 2015
MathSciNet review:
3235184
Full-text PDF Free Access
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Additional Information
Abstract: We consider a stochastic delay differential equation driven by a Hölder continuous process $Z$ and a Wiener process. Under fairly general assumptions on coefficients of the equation, we prove that it has a unique solution. We also give a sufficient condition for finiteness of moments of the solution and prove that the solution depends on $Z$ continuously.
References
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- João Guerra and David Nualart, Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Stoch. Anal. Appl. 26 (2008), no. 5, 1053–1075. MR 2440915, DOI https://doi.org/10.1080/07362990802286483
- K. Kubilius, The existence and uniqueness of the solution of an integral equation driven by a $p$-semimartingale of special type, Stochastic Process. Appl. 98 (2002), no. 2, 289–315. MR 1887537, DOI https://doi.org/10.1016/S0304-4149%2801%2900145-4
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- Yulia S. Mishura and Georgiy M. Shevchenko, Existence and uniqueness of the solution of stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index $H>1/2$, Comm. Statist. Theory Methods 40 (2011), no. 19-20, 3492–3508. MR 2860753, DOI https://doi.org/10.1080/03610926.2011.581174
- Yuliya Mishura and Georgiy Shevchenko, Mixed stochastic differential equations with long-range dependence: Existence, uniqueness and convergence of solutions, Comput. Math. Appl. 64 (2012), no. 10, 3217–3227. MR 2989350, DOI https://doi.org/10.1016/j.camwa.2012.03.061
- Salah-Eldin A. Mohammed, Stochastic differential systems with memory: theory, examples and applications, Stochastic analysis and related topics, VI (Geilo, 1996) Progr. Probab., vol. 42, Birkhäuser Boston, Boston, MA, 1998, pp. 1–77. MR 1652338
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- David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
- G. M. Shevchenko, Integrability of solutions to mixed stochastic differential equations, Ukr. Math. Bulletin 10 (2013), no. 4, 559–574.
- Georgiy Shevchenko, Mixed fractional stochastic differential equations with jumps, Stochastics 86 (2014), no. 2, 203–217. MR 3180033, DOI https://doi.org/10.1080/17442508.2013.774404
- Georgiy Shevchenko and Taras Shalaiko, Malliavin regularity of solutions to mixed stochastic differential equations, Statist. Probab. Lett. 83 (2013), no. 12, 2638–2646. MR 3118207, DOI https://doi.org/10.1016/j.spl.2013.08.013
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References
- M. Besalú and C. Rovira, Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion, Bernoulli 18 (2012), no. 1, 24–45. MR 2888697
- B. Boufoussi and S. Hajji, Functional differential equations driven by a fractional Brownian motion, Comput. Math. Appl. 62 (2011), no. 2, 746–754. MR 2817911 (2012e:60157)
- L. H. Duc, B. Schmalfuss, and S. Siegmund, Generation of random dynamical systems from fractional stochastic delay differential equations, arXiv:math.DS/1309.6478 (2013).
- M. Ferrante and C. Rovira, Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter $H>\frac 12$, Bernoulli 12 (2006), no. 1, 85–100. MR 2202322 (2007b:60144)
- M. Ferrante and C. Rovira, Convergence of delay differential equations driven by fractional Brownian motion, J. Evol. Equ. 10 (2010), no. 4, 761–783. MR 2737158 (2011k:60200)
- J. Guerra and D. Nualart, Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Stoch. Anal. Appl. 26 (2008), no. 5, 1053–1075. MR 2440915 (2009k:60124)
- K. Kubilius, The existence and uniqueness of the solution of an integral equation driven by a $p$-semimartingale of special type, Stochastic Process. Appl. 98 (2002), no. 2, 289–315. MR 1887537 (2003a:60082)
- J. A. León and S Tindel, Malliavin calculus for fractional delay equations, J. Theoret. Probab. 25 (2012), no. 3, 854–889. MR 2956216
- X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Horwood Publishing, Chichester, 2007. MR 2380366 (2009e:60004)
- Yu. S. Mishura and G. M. Shevchenko, Stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index $H>\frac 12$, Comm. Statist. Theory Methods 40 (2011), no. 19–20, 3492–3508. MR 2860753 (2012m:60124)
- Yu. S. Mishura and G. M. Shevchenko, Mixed stochastic differential equations with long-range dependence: Existence, uniqueness and convergence of solutions, Comput. Math. Appl. 64 (2012), no. 10, 3217–3227. MR 2989350
- S.-E. A. Mohammed, Stochastic differential systems with memory: theory, examples and applications, Stochastic Analysis and Related Topics, VI (Geilo, 1996), Progr. Probab., vol. 42, Birkhäuser Boston, Boston, MA, 1998, pp. 1–77. MR 1652338 (99k:60155)
- A. Neuenkirch, I. Nourdin, and S. Tindel, Delay equations driven by rough paths, Electron. J. Probab. 13 (2008), 2031–2068. MR 2453555 (2010b:60115)
- D. Nualart and A. Răşcanu, \text{Differential equations driven by fractional Brownian motion}, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308 (2003f:60105)
- G. M. Shevchenko, Integrability of solutions to mixed stochastic differential equations, Ukr. Math. Bulletin 10 (2013), no. 4, 559–574.
- G. M. Shevchenko, Mixed fractional stochastic differential equations with jumps, Stochastics 86 (2014), no. 2, 203–217. MR 3180033
- G. M. Shevchenko and T. O. Shalaiko, Malliavin regularity of solutions to mixed stochastic differential equations, Stat. Probab. Letters 83 (2013), no. 12, 2638–2646. MR 3118207
- M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Relat. Fields 111 (1998), no. 3, 333–374. MR 1640795 (99j:60073)
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Additional Information
G. Shevchenko
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, Ukraine
Email:
zhora@univ.kiev.ua
Keywords:
Fractional Brownian motion,
Wiener process,
stochastic delay differential equation,
mixed stochastic differential equation
Received by editor(s):
February 26, 2013
Published electronically:
January 26, 2015
Additional Notes:
The research was partially supported by the President’s Grant for Young Scientists, Project GP/F49/002
Article copyright:
© Copyright 2015
American Mathematical Society