On construction of Markov processes
Let σ be the time of hitting 0 for a standard Markov process X(t) with state space (0, ∞), probabilities Px and sub-probability operators Ptf(x)=Ex(f(X(t)); t<σ). An extension of X(t) is a Markov process whose induced probabilities ¯Px satisfy Ex(f(X(t)); t<σ)=Ptf(x). Excursion theory yields for each extension an entrance law ηs, s>0 (ηsP t =ηt+s). Conversely given ηs, one may construct an excursion (or characteristic) measure on path space and a Poisson point process Y. K. Ito and S. Watanabe refer to the possibility of using Y to construct an extension having the prescribed entrance law. In this paper we verify that under quite general conditions this is indeed possible.
KeywordsState Space Stochastic Process General Condition Probability Theory Markov Process
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