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 $\int\psi^z{d}S.$ 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}dS_s, t \geq0,$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2024
- DOI:
- 10.48550/arXiv.2403.13791
- arXiv:
- arXiv:2403.13791
- Bibcode:
- 2024arXiv240313791C
- Keywords:
-
- Mathematics - Probability;
- Quantitative Finance - Mathematical Finance;
- 60H05;
- 28B05;
- 60G48