Stochastic Volterra equation driven by Wiener process and fractional Brownian motion. (English) Zbl 1308.60083

Wang, Zhi; Yan, Litan

Abstr. Appl. Anal. 2013, Article ID 579013, 8 p. (2013).

In this paper, a class of so-called mixed stochastic Volterra equations on \(\mathbb{R}^d\) is studied. The equations considered are driven by both a Wiener process and fractional Brownian motion. The paper is a continuation and extension of several papers published since 2002.
The main result of the paper, Theorem 1, gives sufficient conditions under which the studied equations have a unique strong solution. The proof of the main theorem is based on the classical Yamada and Watanabe theorem [T. Yamada and S. Watanabe, J. Math. Kyoto Univ. 11, 155–167 (1971; Zbl 0236.60037); ibid. 11, 553–563 (1971; Zbl 0229.60039)]. The proof of Theorem 1 is clearly written in detail. Moreover, the authors derive several auxiliary estimates for the integrals involved in the considered equations. In my opinion, these integral estimates, formulated in Propositons 2–7, may be useful for other authors studying mixed stochastic equations.
The paper finishes with references which are representative for the studied problems.

Reviewer: Anna Karczewska (Zielona Gora)

MSC:

60H20	Stochastic integral equations
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60G22	Fractional processes, including fractional Brownian motion

Keywords:

mixed stochastic Volterra equations; Wiener process; fractional Brownian motion

Citations:

Zbl 0236.60037; Zbl 0229.60039

References:

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