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Matrix calculations for moments of Markov processes

Part of: Groups and semigroups of linear operators, their generalizations and applications Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  02 September 2022

Andrew Daw Open the ORCID record for Andrew Daw [Opens in a new window]  and
Jamol Pender Open the ORCID record for Jamol Pender [Opens in a new window]
Andrew Daw*
Affiliation:
University of Southern California
Jamol Pender*
Affiliation:
Cornell University
*
*Postal address: Marshall School of Business, University of Southern California, 401B Bridge Hall, Los Angeles, CA 90089. Email address: andrew.daw@usc.edu
**Postal address: School of Operations Research and Information Engineering, Cornell University, 228 Rhodes Hall, Ithaca, NY 14853.
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Abstract

Matryoshka dolls, the traditional Russian nesting figurines, are known worldwide for each doll’s encapsulation of a sequence of smaller dolls. In this paper, we exploit the structure of a new sequence of nested matrices we call matryoshkan matrices in order to compute the moments of the one-dimensional polynomial processes, a large class of Markov processes. We characterize the salient properties of matryoshkan matrices that allow us to compute these moments in closed form at a specific time without computing the entire path of the process. This simplifies the computation of the polynomial process moments significantly. Through our method, we derive explicit expressions for both transient and steady-state moments of this class of Markov processes. We demonstrate the applicability of this method through explicit examples such as shot noise processes, growth–collapse processes, ephemerally self-exciting processes, and affine stochastic differential equations from the finance literature. We also show that we can derive explicit expressions for the self-exciting Hawkes process, for which finding closed-form moment expressions has been an open problem since their introduction in 1971. In general, our techniques can be used for any Markov process for which the infinitesimal generator of an arbitrary polynomial is itself a polynomial of equal or lower order.

Keywords

Markov processesmatryoshkan matricesmomentspolynomial processes

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 47D07: Markov semigroups and applications to diffusion processes 65C40: Computational Markov chains
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 126 - 150
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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