Existence and uniqueness of the solution of a stochastic differential equation, driven by fractional Brownian motion with a stabilizing term

Authors:
Yu. S. Mishura and S. V. Posashkov

Translated by:
O. I. Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **76** (2007).

Journal:
Theor. Probability and Math. Statist. **76** (2008), 131-139

MSC (2000):
Primary 60G15; Secondary 60H05, 60H10

Published electronically:
July 16, 2008

MathSciNet review:
2368745

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A stochastic differential equation driven by a Wiener process and fractional Brownian motion is considered in the paper. We prove the existence and uniqueness of the solution if the equation contains a certain stabilizing term.

**1.**David Nualart and Aurel Răşcanu,*Differential equations driven by fractional Brownian motion*, Collect. Math.**53**(2002), no. 1, 55–81. MR**1893308****2.**Yuriy Krvavych and Yuliya Mishura,*Exponential formula and Girsanov theorem for mixed semilinear stochastic differential equations*, Mathematical finance (Konstanz, 2000) Trends Math., Birkhäuser, Basel, 2001, pp. 230–238. MR**1882834****3.**P. Cheridito,*Regularizing Fractional Brownian Motion with a View towards Stock Price Modelling*, Ph.D. Thesis, Swiss Federal Institute of Technology, Zurich, 2001.**4.**Masuyuki Hitsuda,*Representation of Gaussian processes equivalent to Wiener process*, Osaka J. Math.**5**(1968), 299–312. MR**0243614****5.**I. I . Gikhman and A. V. Skorokhod,*Stokhasticheskie differentsialnye uravneniya i ikh prilozheniya*, “Naukova Dumka”, Kiev, 1982 (Russian). MR**678374**

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Additional Information

**Yu. S. Mishura**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Email:
myus@univ.kiev.ua

**S. V. Posashkov**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Email:
corlagon@univ.kiev.ua

DOI:
http://dx.doi.org/10.1090/S0094-9000-08-00737-0

Keywords:
Stochastic differential equation,
existence and uniqueness of the solution,
fractional Brownian motion

Received by editor(s):
December 1, 2005

Published electronically:
July 16, 2008

Additional Notes:
The research of the first coauthor is partially supported by the NATO grant PST.CLG 890408

Article copyright:
© Copyright 2008
American Mathematical Society