Markovian Bridges: Construction, Palm Interpretation, and Splicing

Abstract

By a Markovian bridge we mean a process obtained by conditioning a Markov process X to start in some state x at time 0 and arrive at some state z at time t . Once the definition is made precise, we call this process the (x , t , z)-bridge derived from X . Important examples are provided by Brownian and Bessel bridges, which have been extensively studied and find numerous applications. See for example [ PY1 , SW , Sa , H , EL , AP , BP ].It is part of Markovian folklore that the right way to define bridges in some generality is by a suitable Doob h -transform of the space-time process. This method was used by Getoor and Sharpe [GS4] for excursion bridges, and by Salminen [Sa] for one-dimensional diffusions, but the idea of using h-transforms to construct bridges seems to be much older. Our first object in this paper is to make this definition of bridges precise in a suitable degree of generality, with the aim of dispelling all doubts about the existence of clearly defined bridges for nice Markov processes. This we undertake in Section 2. In Section 3 we establish a conditioning formula involving bridges and continuous additive functionals of the Markov process. This formula can be found in [RY, Ex. (1.16) of Ch. X, p.378] under rather stringent continuity conditions. One of our goals here is to prove the formula in its “natural” setting. We apply the conditioning formula in Section 4 to show how Markovian bridges are involved in a family of Palm distributions associated with continuous additive functionals of the Markov process. This generalizes an approach to bridges suggested in a particular case by Kallenberg [K1], and connects this approach to the more conventional definition of bridges adopted here.