Homogeneity and the Strong Markov Property

The Strong Markov Property

Throughout, X := {X ï¿¿ } ï¿¿≥0 denotes a Lévy process on R ï¿¿ with triple (ï¿¿ ï¿¿ σï¿¿ ï¿¿), and exponent Ψ. And from now on, we let {ï¿¿ ï¿¿ } ï¿¿≥0 denote the natural filtration of X, all the time remembering that, in accord with our earlier convention, {ï¿¿ ï¿¿ } ï¿¿≥0 satisfies the usual conditions. Definition 1. The transition measures of X are the probability measures P ï¿¿ (ï¿¿ ï¿¿ A)...

The Strong Markov Property of the Brownian Motion

Strong Markov property of Poisson processes and Slivnyak formula

We discuss strong Markov property of Poisson point processes and the related stopping sets. Viewing Poisson process as a set indexed random field, we demonstrate how the martingale technique applies to establish the analogues of the classical results: Doob’s theorem, Wald identity in this multi-dimensional setting. In particular, we show that the famous Slivnyak-Mecke theorem characterising the...

Strong Ratio Limit Property for P-recurrent Markov Chains

for some positive integer d, then SRLP holds. It is also shown that the inequality (1.2) is always satisfied in a reversible chain. When P = 1, the results given here reduce to those of Orey [7] for recurrent chains, while for P> 1 they imply SRLP for a class of transient chains. (The definition of SRLP given by Orey required that 7=1 and that r< be independent of i since he was dealing with re...

Annotations of an Example That Markov Process Has No Strong Markov Property

In this paper, we discuss an incorrect example that a Markov process does not satisfy strong Markov property, and analyzes the reason of mistake. In the end, we point out it is not reasonable to define strong Markov property by using transition probability functions since transition probability functions might not be one and only.