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A time fractional functional differential equations driven by the fractional Brownian motion
Keywords:Fractional Brownian motion; The Caputo derivative; Stochastic functional differential equation; time delay
Abstract:
      Let $B^H$ be a fractional Brownian motion with Hurst index $H>\frac12$. In this paper, we prove the global existence and uniqueness of the equation $$ \begin{cases} ^CD_t^{\gamma}X(t)=f(x_t)dt+G(x_t)dB^H(t), & t\in (0,T] \\ X(t)=\eta(t), & t\in [-r,0], \end{cases} $$ where $\max\{H,2-2H\}<\gamma<1$, $^CD_t^{\gamma}$ is the Caputo derivative, and $x_t\in \mathcal{C}_r=\mathcal{C}([-r,0],\mathbb{R})$ with $x_t(u)=x(t+u),u\in[-r,0]$. We also study the dependence of the solution on the initial condition.