Markov bridge processes

\[ \renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}} \renewcommand{\disteq}{\stackrel{d}{=}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\vrv}[1]{\vv{\rv{#1}}} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}} \]

A bridge process for some time-indexed Markov process \(\{\rv{x}(t)\}_{t\in[0,T]}\) is obtained from that process by conditioning it to attain a fixed value at the final time \(\rv{x}(T)=Y\) starting from \(\rv{x}(0)=X\), on an interval \([0,T]\). We write that as \(\{\rv{x}(t)\mid \rv{x}(0)=X,\rv{x}(T)=Y\}_{t\in[0,T]}.\) 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.

There is a more general version given in e.g. Privault and Zambrini (2004) where the initial and final states of the process are permitted to have distributions rather than fixed values. That distribution→distribution variant has recently become popular in generative modelling because although it is not easy to solve analytically, it is easy to learn; see Learning Schrödinger Bridges.

This notebook is about the most basic version of the bridge process.

1 Lévy bridges

I am mostly interested in bridges for Lévy increment processes in particular.

It is easy to define a bridge for a Lévy process. For simplicity, we shall take it to be \(\{\rv{x}(t)\}_{t\in [0,1]}\) for \(s\in(0,1)\) and assume \(\rv{x}(0)=1\). Suppose we wish to find the marginal distribution of one point on the bridge \(\{\rv{x}(s)\mid \rv{x}(1)=Y\}_{s\in(0,1)}.\) We know \(\rv{x}(1)\disteq \rv{x}'(s)+\rv{x}''(1-s)\) where \(\rv{x}'(s)\) is an independent copy of \(\rv{x}\). Also, it follows that \(\Ex\rv{x}''(1-s) = (1-s)\Ex\rv{x}(1)\), \(\Ex\rv{x}'(s) = s\Ex\rv{x}(1)\) and \(\left(\rv{x}(s)\mid \rv{x}(1)=Y\right)+\rv{x}'''(1-s)\disteq \rv{x}(1).\) The bridge is a random function mapping \(y\) into the required conditional. TBC

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. There is an introduction for Lévy processes in (Bertoin 1996, VIII.3). 🚧TODO🚧 clarify

Fitzsimmons, Pitman, and Yor (1993) describes a general assortment of methods for handling the properties of bridges, and Perman, Pitman, and Yor (1992) specialises them on pure jump processes (Poisson, gamma).

🏗

1.1 When is a Lévy bridge distribution computationally tractable?

I know it is simple for Gamma, Brownian, and Poisson Lévy processes. Are there others? Yor (2007) asserts that among the Lévy processes, only Brownian and Gamma processes have closed form expressions for the bridge. That is not right; a Poisson process has a simple bridge (binomial sub-division of the increment). Also, both Gamma and Brownian can admit a deterministic drift term.

Anyway, there is a lot of work on conditioning Brownian processes and more generally Gaussian processes; see Gaussian processes, esp. Gaussian processes with pathwise conditioning.

However, I cannot think of other easy examples. Compound Poisson processes, for example, will have a nasty integral term over an unbounded number of dimensions if we do it the obvious way.

Surely there are proofs about this? I have half a mind to see if I cannot find a classification of processes admitting tractable bridges with my rudimentary variational calculus skills.