DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

MODULATED POISSON MEASURES ON ABSTRACT SPACES
JEWGENI H. DSHALALOW
We introduce a notion of a random measure ξ whoseparameterschangeinaccordance
with the evolution of a stochastic process η. Such a measure is called an η-modulated
random measure. A class of problems like this stems from stochastic control theory, but
in the present paper we are more focused on various constructions of modulated random
measures on abstract spaces as well as the formation of functionals of a random process
η with respect to measurable functions and an η-modulated random measure (so-called
potentials), specifically applied to the class of η-modulated marked Poisson random measures.
Copyright © 2006 Jewgeni H. Dshalalow. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
This paper deals with a formalism of modulated random measures that stem from core
applications in physical sciences, engineering and technology, and applied probability
[5, 10, 11]. One of the typical models is a stock market being constantly perturbed by
economic news, random cataclysms and disasters, including famine, earthquakes, hurricanes,
and political events and wars. This causes the main parameters of stocks or mutual
funds, as well as major indexes to alter dependent on these events. We can think of the
stock market as a stochastic process (such as Brownian motion) modulated by some other
“external” stochastic process that takes values in some space and moving randomly from
state to state. The parameters of stock market will remain homogeneous as long as the
external process sojourns in a set. Once it moves on to another set, the parameters of the
stock market change.
One of the widely accepted forms of modulated processes in the literature is found in
Markov-modulated Poisson processes, in which a Poisson process alters its rate in accordance
with an external Markov chain with continuous time parameter. It goes back to
at least 1977 or even earlier in one of the seminal Neuts’ articles (cf. [9]) and it is still
a very popular topic in queueing known under batch Markov arrival processes. A main
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Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 373–381