New Stochastic Fubini Theorems

Abstract

The classic stochastic Fubini theorem says that if one stochastically integrates with respect to a semimartingale $S$ an $\eta (dz)$-mixture of $z$-parametrized integrands ${\psi }^z$, the result is just the $\eta (dz)$-mixture of the individual $z$-parametrized stochastic integrals $\int {\psi }^{z}dS.$ But if one wants to use such a result for the study of Volterra semimartingales of the form $X_t={\int }_0^t{\Psi }_{t,s}d{S}_s,t\ge 0,$ the classic assumption that one has a fixed measure $\eta$ is too restrictive; the mixture over the integrands needs to be taken instead with respect to a stochastic kernel on the parameter space. To handle that situation and prove a corresponding new stochastic Fubini theorem, we introduce a new notion of measure-valued stochastic integration with respect to a general multidimensional semimartingale. As an application, we show how this allows to handle a class of quite general stochastic Volterra semimartingales.