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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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Bertoin, Jean. 1996. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge ; New York: Cambridge University Press.
Chung, Kai Lai, and John B. Walsh, eds. 2005. “H-Transforms.” In Markov Processes, Brownian Motion, and Time Symmetry, 320–35. Grundlehren Der Mathematischen Wissenschaften. New York, NY: Springer. https://doi.org/10.1007/0-387-28696-9_11.
Chung, Kai Lai, John B. Walsh, and Kai Lai Chung. 2005. Markov Processes, Brownian Motion, and Time Symmetry. 2nd ed. Grundlehren Der Mathematischen Wissenschaften 249. Berlin ; New York: Springer.
Connell, Neil O. 2003. “Conditioned Random Walks and the RSK Correspondence.” Journal of Physics A: Mathematical and General 36 (12): 3049–66. https://doi.org/10.1088/0305-4470/36/12/312.
Dembo, Amir. 2013. “Stochastic Processes Lecture Notes.” http://statweb.stanford.edu/~adembo/math-136/nnotes.pdf.
Doob, J. L. 1957. “Conditional Brownian Motion and the Boundary Limits of Harmonic Functions.” Bulletin de La Société Mathématique de France 85: 431–58. https://eudml.org/doc/urn:eudml:doc:86928.
Émery, Michel, and Marc Yor. 2004. “A Parallel Between Brownian Bridges and Gamma Bridges.” Publications of the Research Institute for Mathematical Sciences 40 (3, 3): 669–88. https://doi.org/10.2977/prims/1145475488.
Fitzsimmons, Pat, Jim Pitman, and Marc Yor. 1993. “Markovian Bridges: Construction, Palm Interpretation, and Splicing.” In 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. https://doi.org/10.1007/978-1-4612-0339-1_5.
Jacod, Jean, and Philip Protter. 1988. “Time Reversal on Levy Processes.” The Annals of Probability 16 (2): 620–41. https://doi.org/10.1214/aop/1176991776.
Levin, David Asher, Y. Peres, and Elizabeth L. Wilmer. 2009. Markov Chains and Mixing Times. Providence, R.I: American Mathematical Society.
O’Connell, Neil. 2003. “A Path-Transformation for Random Walks and the Robinson-Schensted Correspondence.” Transactions of the American Mathematical Society 355 (9): 3669–97. https://doi.org/10.1090/S0002-9947-03-03226-4.
Perman, Mihael, Jim Pitman, and Marc Yor. 1992. “Size-Biased Sampling of Poisson Point Processes and Excursions.” Probability Theory and Related Fields 92 (1, 1): 21–39. https://doi.org/10.1007/BF01205234.
Privault, Nicolas, and Jean-Claude Zambrini. 2004. “Markovian Bridges and Reversible Diffusion Processes with Jumps.” Annales de L’Institut Henri Poincare (B) Probability and Statistics 40 (5, 5): 599–633. https://doi.org/10.1016/j.anihpb.2003.08.001.