A new stochastic Fubini-type theorem : On interchanging expectations and Itô integrals

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Baumann, Michael Heinrich:
A new stochastic Fubini-type theorem : On interchanging expectations and Itô integrals.
In: Sankhya A. (20 März 2020) . - S. 1-13.
ISSN 0976-8378
DOI: https://doi.org/10.1007/s13171-019-00195-y

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Abstract

When a stochastic process is given through an Itô 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 Itô 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.