\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\]

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) specialises them on pure jump processes (Poisson, gamma). Yor (2007) asserts that among the Lévy processes, only Brownian and Gamma processes have closed form expressions for the bridge. That is not right; a Poisson process has a trivial bridge (binomial sub-division of the increment). However, I can imagine it is not trivial more broadly. Compoun Poisson processes, for example, I think will have a combinatorial explosion of terms in general.

🏗

## References

*Lévy Processes*. Cambridge Tracts in Mathematics 121. Cambridge ; New York: Cambridge University Press.

*Markov Processes, Brownian Motion, and Time Symmetry*, 320–35. Grundlehren Der Mathematischen Wissenschaften. New York, NY: Springer.

*Markov Processes, Brownian Motion, and Time Symmetry*. 2nd ed. Grundlehren Der Mathematischen Wissenschaften 249. Berlin ; New York: Springer.

*Journal of Physics A: Mathematical and General*36 (12): 3049–66.

*Bulletin de la Société Mathématique de France*85: 431–58.

*Advances in Applied Mathematics*20 (3): 285–99.

*Publications of the Research Institute for Mathematical Sciences*40 (3): 669–88.

*Seminar on Stochastic Processes, 1992*, edited by E. Çinlar, K. L. Chung, M. J. Sharpe, R. F. Bass, and K. Burdzy, 101–34. Progress in Probability. Boston, MA: Birkhäuser Boston.

*The Annals of Probability*16 (2): 620–41.

*Markov Chains and Mixing Times*. Providence, R.I: American Mathematical Society.

*Transactions of the American Mathematical Society*355 (9): 3669–97.

*Probability Theory and Related Fields*92 (1): 21–39.

*Annales de l’Institut Henri Poincare (B) Probability and Statistics*40 (5): 599–633.

*Advances in Mathematical Finance*, edited by Michael C. Fu, Robert A. Jarrow, Ju-Yi J. Yen, and Robert J. Elliott, 37–47. Applied and Numerical Harmonic Analysis. Birkhäuser Boston.

There is a more general version given in (Privault and Zambrini 2004) where the initial and final states of the process are permitted to have distributions rather than fixed values.↩︎

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