On some stochastic differential equations and fractional Brownian motion

Khairia El-Said Abd El-Fattah El-Nadi Alexandria University, Egypt

Let {B_{t}^{H}, t ∈ [0, 1]} be a fractional Brownian motion with Hurst parameter H. In this paper, we proved the existence and uniqueness of strong solution for a stochastic partial differential equation of the following type

u(x, t) = u_{o}(x) + B_{t}^{H} + ∫_{0}^{t} ∑_{|q|≤2m} a_{q}(x)D^{q}u(x, s)ds + ∫_{0}^{t} f(x, t, s, u(x, s))ds,

where D^{q} = D_{1}^{q_{1}}...D_{n}^{q_{n}}, and D_{j} = \frac{∂}{∂x_{j}} and |q| = q_{1}+q_{2}+...+q_{n}, q = (q_{1}, q_{2}, ..., q_{n}) is an n-dimensional multi-index. It is assumed that the operator \frac{∂}{∂t} −∑_{|q|=2m} a_{q}(x)D^{q} is uniformly parabolic.