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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 D. Nualart and 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.

MSC:

60H20 Stochastic integral equations
60G22 Fractional processes, including fractional Brownian motion
60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus

Citations:

Zbl 1018.60057
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References:

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