State spaces for Markov chains

Abstract

If p ( t , i , j ) p(t,i,j) is the transition probability (from i to j in time t) of a continuous parameter Markov chain, with p ( 0 + , i , i ) = 1 p(0 + ,i,i) = 1 , entrance and exit spaces for p are defined. If L [ L ∗ ] L[{L^ \ast }] is an entrance [exit] space, the function p ( ⋅ , ⋅ , j ) [ p ( ⋅ , i , ⋅ ) / h ( ⋅ ) ] p( \cdot , \cdot ,j)[p( \cdot ,i, \cdot )/h( \cdot )] has a continuous extension to ( 0 , ∞ ) × L [ ( 0 , ∞ ) × L ∗ (0,\infty ) \times L[(0,\infty ) \times {L^ \ast } , for a certain norming function h on L ∗ {L^ \ast } ]. It is shown that there is always a space which is both an entrance and exit space. On this space one can define right continuous strong Markov processes, for the parameter interval [0, b], with the given transition function as conditioned by specification of the sample function limits at 0 and b.