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

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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.**D. Nualart and A. Răşcanu,*Differential equation driven by fractional Brownian motion*, Collect. Math.**53**(2002), no. 1, 55-81. MR**1893308 (2003f:60105)****2.**Yu. S. Mishura,*Stochastic Calculus for Fractional Brownian Motion and Related Processes*, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2008. MR**2378138 (2008m:60064)****3.**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)****4.**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)****5.**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

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