Quasi-linear stochastic differential equations with fractional Brownian component. (Ukrainian, English) Zbl 1050.60060

Mishura, Yu. S.

Teor. Jmovirn. Mat. Stat. 68, 95-106 (2003); translation in Theory Probab. Math. Stat. 68, 103-116 (2004).

The author considers stochastic differential equations with fractional Brownian motion. It is proved that fractional Brownian motion belongs to the Hida space \(S^{*}\). Conditions on the stochastic process \(Y\) under which the Wick product \(Y\diamond W_{H}\) with fractional Brownian motion \(W_{H}\) is \(S^{*}\)-integrable are obtained. The author proves that \(S^{*}\)-integral with non-random integrand and “direct” fractional Brownian motion is reduced to ordinary stochastic Itô’s integral with respect to the Wiener process. Two approaches for solving stochastic differential equations with fractional Brownian motion are considered. The first one is based on the Lipschitz condition and a condition for negative norms of coefficients. The second approach is applied only to quasi-linear stochastic differential equations and based on the Giessing lemma.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

quasi-linear stochastic differential equations; fractional Brownian component; existence and uniqueness theorem