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Stochastic Volterra equations driven by fractional Brownian motion with Hurst parameter $H>1/2$. (English) Zbl 1261.60066
Consider the following stochastic Volterra equation in $\mathbb{R}^d$, $$ X(t) = X_0 + \int_0^t b(t,s,X(s))\,ds + \int_0^t \sigma(t,s,X(s))dW^H_s, \ t \in [0,T], $$ where $W^{H,j}_t$ for $j=1,\dotsc,m$ are independent fractional Brownian motions with Hurst parameter $H > \frac{1}{2}$. In this paper, the authors extend results by {\it D. Nualart} and {\it A. Răşcanu} [“Differential equations driven by fractional Brownian motion”, Collect. Math. 53, No. 1, 55--81 (2002; Zbl 1018.60057)], to multidimensional stochastic Volterra equations. Using the Riemann-Stieltjes integral, they give the existence and uniqueness of a solution to the above equation. They also show that the solution has finite moments. Their main aim is to obtain precise estimates for Lebesgue and Riemann-Stieltjes Volterra integrals.

60H20Stochastic integral equations
60G22Fractional processes, including fractional Brownian motion
60H05Stochastic integrals
60H07Stochastic calculus of variations and the Malliavin calculus
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