On solutions of Kolmogorov's equations for jump Markov processes

2013/01/29 by Eugene A. Feinberg, Feinberg, Eugene A., Manasa Mandava +3 · 1 citation
Mathematics · #60J25 #90C40 #FOS: Mathematics #Mathematical Dynamics and Fractals #Morphological variations and asymmetry #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1301.6998

openalex publication_date 2013/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Feller's seminal paper. In particular, this paper extends Feller's results for continuous Q-functions to measurable Q-functions and provides additional results.

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