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.
Similar content being viewed by others
References
Barmish, B.R. and Primbs, J.A. (2011). On arbitrage possibilities via linear feedback in an idealized brownian motion stock market. IEEE, p. 2889–2894.
Baumann, M.H. (2017). On stock trading via feedback control when underlying stock returns are discontinuous. IEEE Trans. Automat. Control 62, 6, 2987–2992.
Baumann, M.H. (2018). Performance and effects of linear feedback stock trading strategies. doctoral thesis, Universität Bayreuth.
Berger, M.A. and Mizel, V.J. (1979). Theorems of Fubini type for iterated stochastic integrals. Trans. Amer. Math. Soc. 252, 249–274.
Hille, T. (2014). On a weak* stochastic Fubini theorem. ETH Zürich, pp. 1–55.
Kailath, T., Segall, A. and Zakai, M. (1978). Fubini-type theorems for stochastic integrals. Sankhyā: The Indian Journal of Statistics Series A 40, 2, 138–143.
Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 1–2, 125–144.
Protter, P.E. (2005). Stochastic Integration and Differential Equations. Springer.
van Neerven, J. and Veraar, M.C. (2005). On the stochastic Fubini theorem in infinite dimensions. CRC Press, Citeseer, p. 323–336.
Veraar, M.C. (2012). The stochastic Fubini theorem revisited. Stochastics: An International Journal of Probability and Stochastic Processes 84, 4, 543–551.
Wald, A. (1944). On cumulative sums of random variables. Ann. Math. Stat. 15, 3, 283–296. the Institute of Mathematical Statistics.
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).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interests
Conflict of interests. The author declares that he has no conflict of interest.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The work of the author was supported by Hanns-Seidel-Stiftung e. V. (HSS), funded by Bundesministerium für Bildung und Forschung (BMBF).
Rights and permissions
About this article
Cite this article
Baumann, M.H. A New Stochastic Fubini-Type Theorem. Sankhya A 83, 408–420 (2021). https://doi.org/10.1007/s13171-019-00195-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-019-00195-y