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A New Stochastic Fubini-Type Theorem

On Interchanging Expectations and Itō Integrals

Abstract

When a stochastic process is given through an Itō integral, i.e. a stochastic integral, or a stochastic differential equation (SDE), an analytical solution does not have to exist—and even if there is a closed-form solution, the derivation of this solution can be very complex. When the solution of the stochastic process is not needed but only the expected value as a function of time, the question arises whether it is possible to use the expectation operator directly on the stochastic integral or on the SDE and to somehow calculate the expectation of the process as a Riemann integral over the expectation of the integrands and integrators. In this paper, we show that if the integrator is linear in expectation, the expectation operator and an Itō integral can be interchanged. Additionally, we state how this can be used on SDEs and provide an application from the field of technical trading, i.e. from the field of mathematical finance.

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Acknowledgements

The author wishes to thank Michaela Baumann, Lars Grüne, Melanie Birke, and Bernhard Herz. This work is dedicated to Valentin. Parts of this work also appeared in the doctoral thesis of the author (Baumann, 2018).

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Correspondence to Michael Heinrich Baumann.

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The work of the author was supported by Hanns-Seidel-Stiftung e. V. (HSS), funded by Bundesministerium für Bildung und Forschung (BMBF).

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Baumann, M.H. A New Stochastic Fubini-Type Theorem. Sankhya A 83, 408–420 (2021). https://doi.org/10.1007/s13171-019-00195-y

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  • DOI: https://doi.org/10.1007/s13171-019-00195-y

Keywords

  • Stochastic analysis
  • Itō integral
  • Fubini theorem
  • Semimartingale.

AMS (2000) subject classification

  • 60H05
  • 60H10