Markovian bridges: Weak continuity and pathwise constructions

TL;DR

This paper constructs weakly continuous Markovian bridges for Feller processes, extends the strong Markov property with reversed time flow, and applies these results to self-similar processes.

Contribution

It introduces a unique weakly continuous construction of Markovian bridges for Feller processes and extends the strong Markov property with reversed time flow.

Findings

Constructed weakly continuous Markovian bridges for Feller processes

Extended the strong Markov property to include reversed time flow

Applied the construction to self-similar Feller processes

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.