A note on stochastic Fubini's theorem and stochastic convolution

TL;DR

This paper extends stochastic Fubini's theorem to a broader setting, allowing for stochastic convolutions with general operator-valued maps, and establishes conditions for their existence and continuity.

Contribution

It introduces a version of stochastic Fubini's theorem independent of specific integrators and characterizes stochastic convolutions with general operator-valued maps on Hilbert spaces.

Findings

Established a generalized stochastic Fubini's theorem.

Proved existence of predictable and continuous versions of stochastic convolutions.

Characterized measurability conditions for operator-valued processes in convolutions.

Abstract

We provide a version of the stochastic Fubini's theorem which does not depend on the particular stochastic integrator chosen as far as the stochastic integration is built as a continuous linear operator from an $L^p$ space of Banach space-valued processes (the stochastically integrable processes) to an $L^p$ space of Banach space-valued paths (the integrated processes). Then, for integrators on a Hilbert space $H$, we consider stochastic convolutions with respect to a strongly continuous map $R:(0,T]\rightarrow L(H)$, not necessarily a semigroup. We prove existence of predictable versions of stochastic convolutions and we characterize the measurability needed by operator-valued processes in order to be convoluted with $R$. Finally, when $R$ is a $C_0$-semigroup and the stochastic integral provides continuous paths, we show existence of a continuous version of the convolution, by adapting the factorization method to the present setting.