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Stochastic differential equations driven by fractional Brownian motions. (English) Zbl 1214.60024

The authors study the well-posedness of a class of stochastic differential equations driven by fractional Brownian motion with arbitrary Hurst parameter and with random coefficients that are possibly anticipating. The stochastic integral is defined in the Skorokhod sense. The main idea is to establish a generalized version of the anticipating Girsanov theorem in the fBm setting and then to follow a scheme of R. Buckdahn [Probab. Theory Relat. Fields 90, No. 2, 223–240 (1991; Zbl 0735.60057)] to attack the problem of well-posedness. A major component in this method is the generalization of a fundamental theorem by S. Kusuoka [J. Fac. Sci., Univ. Tokyo, Sect. I A 29, 567–597 (1982; Zbl 0525.60050] on anticipating Girsanov transformation. Using all these tools, the existence and uniqueness of the solution of SDE involving fBm is proved. The drift coefficient can be even nonlinear, which is an improvement, even compared to the original result of Buckdahn.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G22 Fractional processes, including fractional Brownian motion
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