New Stochastic Fubini Theorems
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914721629732864 |
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| author | Choulli, Tahir Schweizer, Martin |
| author_facet | Choulli, Tahir Schweizer, Martin |
| contents | 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 $ψ^z$, the result is just the $η(dz)$-mixture of the individual $z$-parametrized stochastic integrals $\intψ^z{d}S.$ But if one wants to use such a result for the study of Volterra semimartingales of the form $ X_t =\int_0^t Ψ_{t,s}dS_s, t \geq0,$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13791 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New Stochastic Fubini Theorems Choulli, Tahir Schweizer, Martin Probability Mathematical Finance 60H05, 28B05, 60G48 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 $ψ^z$, the result is just the $η(dz)$-mixture of the individual $z$-parametrized stochastic integrals $\intψ^z{d}S.$ But if one wants to use such a result for the study of Volterra semimartingales of the form $ X_t =\int_0^t Ψ_{t,s}dS_s, t \geq0,$ 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. |
| title | New Stochastic Fubini Theorems |
| topic | Probability Mathematical Finance 60H05, 28B05, 60G48 |
| url | https://arxiv.org/abs/2403.13791 |