| Volume 9, Number 2, 2019, Pages 547-567 DOI：10.11948/2156-907X.20180068 |
| A time fractional functional differential equation driven by the fractional Brownian motion |
| Jingqi Han,Litan Yan |
| 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
$$
\begin{cases}
^CD_t^{\gamma}x(t)=f(x_t)+G(x_t)\frac{d}{dt}B^H(t),\ \ \ \ &t\in(0,T], \x(t)=\eta(t), \ \ \ \ \ &t\in[-r,0],
\end{cases}
$$
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. |
PDF Download reader
|
|
|
|