Vector-valued stochastic delay equations - a semigroup approach

2010/06/28 by Sonja Cox, Cox, Sonja, Mariusz Górajski +1
Economics, Econometrics and Finance · Engineering · Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations #Stochastic processes and financial applications #math.FA

paper · pdf · doi:10.48550/arxiv.1006.5349

published version; 17 pages, no figures

openalex publication_date 2010/06/28 · arxiv created 2010/11/12 · arxiv updated 2010/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be a type 2 UMD Banach space, H a Hilbert space and let p be in [1,∞). 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) = x₀; X₀ = f₀. Here A : D(A) -> E is the generator of a C₀-semigroup, the operator C is given by a Riemann-Stieltjes integral, B : E x L^p(-1,0;E) -> γ(H,E) is a Lipschitz function and W_H is an H-cylindrical Brownian motion. We prove that a solution to \eqrefSDE1 is equivalent to a solution to the corresponding stochastic Cauchy problem, and use this to prove the existence, uniqueness and continuity of a solution.

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