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
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 83 (2010).
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
Abstract | References | Similar Articles | Additional Information
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.
- 1. David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
- 2. Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- 3. 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
- 4. 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
- 5. M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields 111 (1998), no. 3, 333–374. MR 1640795, https://doi.org/10.1007/s004400050171
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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
DOI:
https://doi.org/10.1090/S0094-9000-2012-00845-4
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
