Continuous dependence of solutions of stochastic differential equations driven by standard and fractional Brownian motion on a parameter
Authors:
Yu. S. Mishura, S. V. Posashkova and S. V. Posashkov
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 83 (2011), 111-126
MSC (2010):
Primary 60G22; Secondary 60H10
DOI:
https://doi.org/10.1090/S0094-9000-2012-00845-4
Published electronically:
February 2, 2012
MathSciNet review:
2768852
Full-text PDF Free Access
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Abstract: We consider a stochastic differential equation driven by both a Wiener process and a fractional Brownian motion. The coefficients of the equation are nonhomogeneous, and the initial condition is random. It is assumed that both the coefficients and the initial condition depend on a parameter. We establish conditions on the coefficients and the initial condition for the continuous dependence of a solution on the parameter.
References
- David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
- Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- Yulia Mishura and Sergiy Posashkov, Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and Wiener process, Theory Stoch. Process. 13 (2007), no. 1-2, 152–165. MR 2343820
- Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
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References
- D. Nualart and A. Răşcanu, Differential equation driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308 (2003f:60105)
- Yu. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2008. MR 2378138 (2008m:60064)
- Yu. S. Mishura and S. V. Posashkov, Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and Wiener process, Theory Stoch. Process. 29 (2007), 152–165. MR 2343820 (2009c:60158)
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. MR 1347689 (96d:26012)
- M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields 111 (1988), no. 3, 333–374. MR 1640795 (99j:60073)
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Additional Information
Yu. S. Mishura
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
myus@univ.kiev.ua
S. V. Posashkova
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
revan1988@gmail.com
S. V. Posashkov
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
corlagon@univ.kiev.ua
Keywords:
Fractional Brownian motion,
standard Brownian motion,
stochastic differential equation,
continuity in a parameter
Received by editor(s):
May 28, 2010
Published electronically:
February 2, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
