Matrix calculations for moments of Markov processes

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