Vector-valued stochastic delay equations—a semigroup approach

Abstract

Abstract. Let E be a type 2 umd Banach space, H a Hilbert space and let p ∈ [1, ∞). Consider the following stochastic delay equation in E:whereis given by a Riemann-Stieltjes integral, and B :We prove that a solution to (SDE) is equivalent to a solution to the corresponding stochastic Cauchy problem, and use this to prove the existence, uniqueness and continuity of a solution to (SDE).

Vector-valued stochastic delay equations—A weak solution and its Markovian representation

“…In the sequel we need Lemma 3.3 (cf. Lemma 4.1 in [9]). The proof of the following lemma is left to the reader.…”

“…A generalised strong solution to (SCP) is defined and its equivalence to a mild solution of (SCP) is proven in [9]. Definition 2.5.…”

“…For a general class of spaces including the E p spaces the variation of constants formula for finite-dimensional delay equations with additive noise and a bounded delay operator is discussed in Riedle [23]. For more references see [9]. We complement and extend result from [9] concerning existence and uniqueness of solution to (SDE) by adding non-linear part and introducing weak concept of solution to (SDE).…”

Stochastic Integration in Banach Spaces – a Survey

“…Applications of the theory of stochastic integration in UMD spaces have been worked out in a number of papers; see [12,13,18,21,22,24,30,57,56,58,78,80,81,93,96] and the references therein. Here we will limit ourselves to the maximal regularity theorem for stochastic convolutions from [81] which is obtained by combining Theorem 7.1 and 7.3 below, and which crucially depends on the sharp two-sided inequality of Theorem 5.5.…”

On the Equivalence of Solutions for a Class of Stochastic Evolution Equations in a Banach Space

“…Hence using the multiplier theorem due to Kalton and Weis [17] we obtain: if G(Y (·)) is in γ(L 2 (0, t; H), E) a.s., then the processes represent elements in γ(L 2 (0, t; H), E) a.s. Hence from Lemma 2.8 in [7] it follows that…”