Regularization of differential equations by fractional noise. (English) Zbl 1075.60536

Nualart, David; Ouknine, Youssef

Stochastic Processes Appl. 102, No. 1, 103-116 (2002).

Existence and uniqueness of a strong solution to the differential equation \[ X_t= x + \int _0^t b(s,X_s)\,ds + B^H_t, \quad t\geq 0, \] is established, where \(B^H_t\) is a fractional Brownian motion with the Hurst parameter \(H\in (0,1)\) and \(b(s,x)\) is a bounded Borel function with at most linear growth in \(x\) (for \(H\leq 1/2\)) or Hölder continuous function of order strictly larger than \(1-1/2H\) in \(x\) and \(H-1/2\) in time.

Reviewer: Bohdan Maslowski (Praha)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60G18	Self-similar stochastic processes

Keywords:

fractional Brownian motion; Malliavin calculus

References:

[1]	Alòs, E.; Mazet, O.; Nualart, D., Stochastic calculus with respect to Gaussian processes, Ann. probab., 29, 766-801, (2001) · Zbl 1015.60047
[2]	Decreusefond, L.; Üstunel, A.S., Stochastic analysis of the fractional Brownian motion, Potential anal., 10, 177-214, (1999) · Zbl 0924.60034
[3]	Fernique, X.M., 1974. Regularité des trajectories de fonctions aléatoires gaussiennes. In: Hennequin, P.L. (Ed.), École d’Eté de Saint-Flour IV, Lecture Notes in Mathematics, Vol. 480, Springer, Berlin, pp. 2-95.
[4]	Friedman, A., Stochastic differential equations and applications, (1975), Academic Press New York
[5]	Gyöngy, I.; Pardoux, E., On quasi-linear stochastic partial differential equations, Probab. theory related fields, 94, 413-425, (1993) · Zbl 0791.60047
[6]	Moret, S., Nualart, D., 2002. Onsager-Machlup functional for the fractional Brownian motion. Probab. Theory Related Fields, to appear. · Zbl 1007.60031
[7]	Nakao, S., On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations, Osaka J. math., 9, 513-518, (1972) · Zbl 0255.60039
[8]	Norros, I.; Valkeila, E.; Virtamo, J., An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motion, Bernoulli, 5, 571-587, (1999) · Zbl 0955.60034
[9]	Ouknine, Y., Généralisation d’un lemme de S. nakao et applications, Stochastics, 23, 149-157, (1988) · Zbl 0637.60059
[10]	Samko, S.G.; Kilbas, A.A.; Mariachev, O.I., Fractional integrals and derivatives, (1993), Gordon and Breach Science London
[11]	Veretennikov, A.Ju., On strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR-sb., 39, 387-403, (1981) · Zbl 0462.60063
[12]	Zähle, M., Integration with respect to fractal functions and stochastic calculus I, Probab. theory related fields, 111, 333-374, (1998) · Zbl 0918.60037
[13]	Zvonkin, A.K., A transformation of the phase space of a diffusion process that removes the drift, Math. USSR-sb., 22, 129-149, (1974) · Zbl 0306.60049