Journal
ANNALS OF OPERATIONS RESEARCH
Volume 317, Issue 2, Pages 587-604Publisher
SPRINGER
DOI: 10.1007/s10479-017-2538-8
Keywords
Jump Markov process; Kolmogorov's equation; Minimal solution; Boundedness condition; Transition function; Unbounded transition rates
Categories
Funding
- National Science Foundation [CMMI-1335296, CMMI-1636193]
- Div Of Civil, Mechanical, & Manufact Inn
- Directorate For Engineering [1335296] Funding Source: National Science Foundation
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This article investigates the solutions of Kolmogorov's backward and forward equations for jump Markov processes. The authors found that the minimal solution is the transition probability if the transition rate is bounded. The paper also presents more general results, providing sufficient conditions for locally integrable or bounded transition rates.
As is well-known, transition probabilities of jump Markov processes satisfy Kolmogorov's backward and forward equations. In the seminal 1940 paper, William Feller investigated solutions of Kolmogorov's equations for jump Markov processes. Recently the authors solved the problem studied by Feller and showed that the minimal solution of Kolmogorov's backward and forward equations is the transition probability of the corresponding jump Markov process if the transition rate at each state is bounded. This paper presents more general results. For Kolmogorov's backward equation, the sufficient condition for the described property of the minimal solution is that the transition rate at each state is locally integrable, and for Kolmogorov's forward equation the corresponding sufficient condition is that the transition rate at each state is locally bounded.
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