The Living Thing / Notebooks :

Markov bridge processes

Usefulness: 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

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Notes towards general bridge processes for Markov processes.

A bridge process for some time-indexed Markov process \(\{\Lambda(t)\}_{t\in\mathbb{R}}\) has the same distribution as the Bridge process conditional upon attaining a fixed value \(\Lambda(1)=\lambda_1\) starting from \(\Lambda(0)=\lambda_0.\)

We might write that as \(\{\Lambda(t)\mid \Lambda(0)=\lambda_0,\Lambda(1)=\lambda_1\}_{t\in[0,1]}.\)

I am mostly interested in this for Lévy processes in particular rather than general Markov ones. 🚧

Refs

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.

Privault, Nicolas, Avenue Michel Crepeau, La Rochelle, Jean-Claude Zambrini, Universidade de Lisboa, and Avenida Gama Pinto. n.d. “Markovian Bridges and Reversible Diffusion Processes with Jumps,” 58.