Markovian bridges: Weak continuity and pathwise constructions

Abstract

A Markovian bridge is a probability measure taken from a disintegration of the law of an initial part of the path of a Markov process given its terminal value. As such, Markovian bridges admit a natural parameterization in terms of the state space of the process. In the context of Feller processes with continuous transition densities, we construct by weak convergence considerations the only versions of Markovian bridges which are weakly continuous with respect to their parameter. We use this weakly continuous construction to provide an extension of the strong Markov property in which the flow of time is reversed. In the context of self-similar Feller process, the last result is shown to be useful in the construction of Markovian bridges out of the trajectories of the original process.

Published in at http://dx.doi.org/10.1214/10-AOP562 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (27)

• Bridges of Lévy processes conditioned to stay positive
• Fractional $P(\varphi {)}_1$-processes and Gibbs measures
• Poisson ensembles of loops of one-dimensional diffusions
• Integrated density of states for Poisson-Schrödinger perturbations of subordinate Brownian motions on the Sierpiński gasket
• Dini derivatives for Exchangeable Increment processes and applications
• A characterisation of the Gaussian free field
• Uniform LAN property of locally stable Lévy process observed at high frequency
• Stochastic Modelling with Randomised Markov Bridges
• On the profile of trees with a given degree sequence
• Jump stochastic differential equations for the characterisation of the Bragg peak in proton beam radiotherapy
• On the Feynman--Kac semigroup for some Markov processes
• Shifting processes with cyclically exchangeable increments at random
• LAN property for families of distributions of solutions to Levy driven SDE's
• Invariance principles for local times in regenerative settings
• The Spine of the Fleming-Viot process driven by Brownian motion
• Second derivative of the log-likelihood in the model given by a Levy driven stochastic differential equations
• Functional limits for "tied down" occupation time processes of infinite ergodic transformations
• The greatest convex minorant of Brownian motion, meander, and bridge
• On the sizes of burnt and fireproof components for fires on a large Cayley tree
• Lévy bridges with random length
• Malliavin calculus approach to statistical inference for Levy driven SDE's
• Lifschitz tail and sausage asymptotics for stable processes in the Poissonian environment on the Sierpinski gasket
• Generalized Gaussian Bridges
• Markov bridges: SDE representation
• Invariance of three-dimensional Bessel bridges in terms of time reversal
• The Impact of Pinning Points on Memorylessness in Lévy Random Bridges
• Entropic turnpike estimates for the kinetic Schrödinger problem