A note on stochastic Fubini's theorem and stochastic convolution

Mauro Rosestolato

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A note on stochastic Fubini's theorem and stochastic convolution

Mauro Rosestolato

Probability · Submitted 20 Jun 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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).

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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.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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).	contribution 0.96	0.92	0.48	asserted 1.00
s1	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.	contribution 0.85	0.69	0.07	asserted 0.90
s2	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 $R$.	result 0.71	0.96	0.18	asserted 1.00
s3	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.	result 0.87	0.95	0.22	asserted 1.00

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Original 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.

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