Markov bridge processes

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A bridge process for some time-indexed Markov process \(\{\Lambda(t)\}_{t\in[S,T]}\) is obtained from that process by conditioning it to attain a fixed value \(\Lambda(T)=Y\) starting from \(\Lambda(S)=X\) on some interval \([S,T]\). We write that as \(\{\Lambda(t)\mid \Lambda(S)=X,\Lambda(T)=Y\}_{t\in[S,T]}.\) Put another way, given the starting and finishing values of a stochastic markov process, I would like to rewind time to find out the values of its path at a midpoint which is “compatible” with the endpoints.¹

I am mostly interested in this for Lévy processes in particular, rather than arbitrary Markov processes. There is an introduction for these in (Bertoin 1996, VIII.3) but I an not sure that it permits processes with jumps.

For a fairly old and simple concept, work in the area skews surprisingly recent (i.e post 1990), given that applications seem to cite the early, gruelling (Doob 1957) as the foundation; perhaps for about 30 years no-one wanted to learn enough potential theory to make it go.

It is trivial (more or less) to derive the properties of the bridge for a Wiener process, so that can be found in every stochastic processes textbook.

Alexandre Hoang Thiery describes Doob’s h-transform method and notes that while it might be good for deducing certain properties of the paths of such a conditioned process it is not necessarily a good way of sampling it.

(Fitzsimmons, Pitman, and Yor 1993) give us a fairly general bunch of methods for handling the properties of bridges, and (Perman, Pitman, and Yor 1992) give similar tools for pure-jump processes (Poisson, gamma).

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