Stochastic Fubini Theorem for Semimartingales in Hilbert Space

Abstract

In this paper we will study the Fubini theorem for stochastic integrals with respect to semimartingales in Hilbert space.Let (Ω, , P) he a probability space, (X, , μ) a measure space, H and G two Hilbert spaces, L(H, G) the space of bounded linear operators from H into G, Z an H-valued semimartingale relative to a given filtration, and φ: X × R+ × Ω → L(H, G) a function such that for each t ∈ R+ the iterated integrals are well-defined (the integrals with respect to μ are Bochner integrals). It is often necessary to have sufficient conditions for the process Y1 to be a version of the process Y2 (e.g. [1], proof of Theorem 2.11).

Fubini theorem for anticipating stochastic integrals in Hilbert space

“…(ii) Note that the conclusion of the theorem is similar to that of Theorem 3.4 in [11] where ~o(x) is {~}-adapted. (3.2) and (3.3) hold.…”

“…In the case where integrands are "adapted" to a given filtration, Le6n [11] established a result which generalizes those of [3], [6], [7], and [12], and in the case where the integrands are real-valued and "anticipating" with respect to the given filtration, Berger [1] used the homogeneous chaos expansion of the integrands to prove a similar result to that of [11].…”

“…The main purpose of this paper is to prove a stochastic Fubini theorem similar to those of [1] and [11] for anticipating integrals with respect to a cylindrical Wiener process in Hilbert space. This anticipating Fubini theorem is applied in the last part of the paper to extend to the anticipating case the results proved in [9] in the adapted case, for the equivalence of solutions of a certain class of stochastic evolution equations in Hilbert space.…”

“…The stochastic Fubini theorem, which has been a useful tool for the analysis of some classes of stochastic differential equations (see, e.g., [2], [3], and [11]), has been studied by several authors. In the case where integrands are "adapted" to a given filtration, Le6n [11] established a result which generalizes those of [3], [6], [7], and [12], and in the case where the integrands are real-valued and "anticipating" with respect to the given filtration, Berger [1] used the homogeneous chaos expansion of the integrands to prove a similar result to that of [11].…”

Existence of Weak Solutions for a Class of Semilinear Stochastic Wave Equations

“…where S ij (t) denotes the (i, j)-th entry of the operator matrix S(t). A further computation (which is elementary, apart of having to appeal to a general stochastic Fubini's theorem such as the one in [22]) shows that a mild solution satisfies the Duhamel's formulation u(t) + Consider the regularized equation…”