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New Stochastic Fubini Theorems

Tahir Choulli, Martin Schweizer

Abstract

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.

New Stochastic Fubini Theorems

Abstract

The classic stochastic Fubini theorem says that if one stochastically integrates with respect to a semimartingale an -mixture of -parametrized integrands , the result is just the -mixture of the individual -parametrized stochastic integrals But if one wants to use such a result for the study of Volterra semimartingales of the form the classic assumption that one has a fixed measure 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.
Paper Structure (13 sections, 10 theorems, 117 equations)

This paper contains 13 sections, 10 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Semimartingales and stochastic integrals
  5. Measure-valued processes
  6. Stochastic integrals
  7. Simple integrands and integrals
  8. Approximation of general integrands
  9. Limits of measure-valued semimartingales
  10. Two new stochastic Fubini theorems
  11. An application to Volterra semimartingales
  12. Connections to the classic case
  13. An illustrative example

Key Result

Lemma 3.1

Fix an $\mathbb{R}^{d}$-valued semimartingale $S$, a control process $V$ for $S$ and a stopping time $\tau$ with $V_{\tau-} \in L^{2}$. For each $\varphi \in \mathcal{E}$, there exists a process $\varphi \bullet S^{\tau-}=\left(\varphi \bullet S_{t}^{\tau-}\right)_{t \geq 0}$ with values in $\mathbb More precisely, (3.2) states that for each $f \in C(\mathcal{K})$, the two processes $\left(\varphi

Theorems & Definitions (31)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Corollary 3.7
  • ...and 21 more