Stochastic Analysis and Applications · 2008 · 62 citations · 2 references
EngineeringMathematicsProbability TheoryStochastic ModelingStochastic ProcessesMarkov ChainsMarkov ProcessesPoint ProcessesStochastic CalculusCompensatorsContinuous Random VariablesFinite Markov ChainsFinite-state Markov ChainsMultivariate Point ProcessesMultivariate Random VariablesPoint Process DependenceStochastic Process Analysis
The paper investigates how multivariate stochastic processes with specified infinitesimal characteristics can be decomposed into univariate components, extending the dependence framework analogous to Sklar’s theorem for random variables. The study seeks conditions on the infinitesimal characteristics that ensure the univariate components of the multivariate process have prescribed laws. The authors define a mark space for the process and characterize each component \(Z_i\) as a right‑continuous, increasing cadlag process. Keywords: compensators, dependence, stochastic processes; Mathematics Subject Classification: 60G5, 60G57, 60G99; supported by NSF Grants 0202851 and 0604789 and Moody's Corporation grant 5‑55411.
Abstract This article is concerned with studying the following problem: Consider a multivariate stochastic process whose law is characterized in terms of some infinitesimal characteristics, such as the infinitesimal generator in case of finite Markov chains. Under what conditions imposed on these infinitesimal characteristics of this multivariate process, the univariate components of the process agree in law with given univariate stochastic processes. Thus, in a sense, we study a stochastic processe' counterpart of the stochastic dependence problem, which in case of real valued random variables is solved in terms of Sklar's theorem. Keywords: CompensatorsDependenceStochastic processesMathematics Subject Classification: 60G5560G5760G99 Notes 1Some authors (cf. Last and Brandt [Citation2]) refer to marked point process as to multivariate point process. This is not the definition that we have in mind, however. 2We slightly abuse the term "mark space" here. 3By definition, process Z i is right-continuous; since it is increasing—it is cad-lag. This research was supported in part by NSF Grants 0202851 and 0604789, and by Moody's Corporation grant 5-55411.
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