New Stochastic Fubini Theorems

The classic stochastic Fubini theorem says that if one stochastically integrates with respect to a semimartingale S an η(dz)-mixture of z-parametrized integrands ψᶻ, the result is just the η(dz)-mixture of the individual z-parametrized stochastic integrals ∫ψᶻdS. But if one wants to use such a result for the study of Volterra semimartingales of the form Xₜ =∫₀ᵗ Ψt,sdSₛ, t ≥0, the classic assumption that one has a fixed measure η 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.