Vector-valued stochastic delay equations - a semigroup approach

Abstract

Let E be a type 2 UMD Banach space, H a Hilbert space and let p be in [1,\infty). Consider the following stochastic delay equation in E: dX(t) = AX(t) + CX_t + b(X(t),X_t)dW_H(t), t>0; X(0) = x_0; X_0 = f_0. Here A : D(A) -> E is the generator of a C_0-semigroup, the operator C is given by a Riemann-Stieltjes integral, B : E x L^p(-1,0;E) -> \gamma(H,E) is a Lipschitz function and W_H is an H-cylindrical Brownian motion. We prove that a solution to \eqref{SDE1} is equivalent to a solution to the corresponding stochastic Cauchy problem, and use this to prove the existence, uniqueness and continuity of a solution.