Stochastic Differential Equations Driven by a Fractional Brownian Motion
نویسنده
چکیده
We study existence, uniqueness and regularity of some sto-chastic diierential equations driven by a fractional Brownian motion of any Hurst index H 2 (0; 1): 1. Introduction Fractional Brownian motion and other longgrange dependent processes are more and more studied because of their potential applications in several elds like telecommunications networks, nance markets, biology and so on The main theoretical problem raised by the fractional Brownian and related processes is that they are not Markovian, even more they are not semii martingales, hence it is somewhat diicult to setup a stochastic calculus with respect to these processes. In particular, there exist several sensible deeni-tions of a stochastic integral with respect to the fractional Brownian mo-In this work we chose to use the approach deened in (De-creusefond & sttnel, 1997) (which lies on the Gaussiannity of the fractional Brownian motion) and to study stochastic diierential equations within this framework. The equations we have to deal with are of Volterra type but our work seems not to be subsumed by previous articles on this subject (see) because our kernel is weakly regular and we are looking for classical solutions and not distributionnvalued ones. The paper is organized as follows. In section 2, we present some preliminary technical results concerning the Malliavin calculus applied to the fractional Brownian motion. Section 3 is devoted to show the main results on existence and uniqueness of solutions of diierential equations driven by a fractional Brownian motion. In section 4, we discuss the absolute continuity of the law of the solution.
منابع مشابه
Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion
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