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On some fractional stochastic delay differential equations. (English) Zbl 1189.60117
Summary: We consider the Cauchy problem for an abstract stochastic delay differential equation driven by fractional Brownian motion with the Hurst parameter \(H>1/2\). We prove the existence and uniqueness for this problem, when the coefficients have enough regularity, the diffusion coefficient is bounded away from zero and the coefficients are smooth functions with bounded derivatives of any order. We prove the theorem by using the convergence of the Picard-Lindelöf iterations in \(L^{2}(\Omega )\) to a solution of this problem which admits a smooth density with respect to Lebesgue’s measure on \(\mathbb{R}\).

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
26A33 Fractional derivatives and integrals
34K37 Functional-differential equations with fractional derivatives
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