Matrix Calculations for Moments of Markov Processes

Abstract

Matryoshka dolls, the traditional Russian nesting figurines, are known world-wide for each doll's encapsulation of a sequence of smaller dolls. In this paper, we identify a large class of Markov process whose moments are easy to compute by exploiting the structure of a new sequence of nested matrices we call Matryoshkhan matrices. We characterize the salient properties of Matryoshkhan matrices that allow us to compute these moments in closed form at a specific time without computing the entire path of the process. This speeds up the computation of the Markov process moments significantly in comparison to traditional differential equation methods, which we demonstrate through numerical experiments. 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, linear birth-death-immigration 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 its 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