Vector-valued stochastic delay equations - a semigroup approach

TL;DR

This paper develops a semigroup approach to analyze vector-valued stochastic delay equations in UMD Banach spaces, establishing existence, uniqueness, and continuity of solutions using stochastic Cauchy problem equivalence.

Contribution

It introduces a novel semigroup framework for stochastic delay equations in Banach spaces, linking solutions to stochastic Cauchy problems.

Findings

Proves equivalence between stochastic delay equations and Cauchy problems.

Establishes existence and uniqueness of solutions.

Demonstrates solution continuity.

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.