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

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.