Matrix calculations for moments of Markov processes. (2nd March 2023)

• Record Type: Journal Article
• Authors: Daw, Andrew; Pender, Jamol
• Is Part Of: Advances in applied probability. Volume 55:Number 1(2023)
• Page Start: 126
• Page End: 150
• Publication Date: 2023-03-02
• Subjects: Markov processes -- matryoshkan matrices -- moments -- polynomial processes; 60J25 -- 47D07 -- 65C40; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2022.8
• Languages: English
• ISSNs: 0001-8678

Abstract

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.