# Time-Homogeneous Diffusions with a Given Marginal at a Random Time

## TL;DR

This paper explicitly constructs a class of time-homogeneous diffusions that, when stopped at an independent exponential time, follow a specified distribution, using three different proof techniques.

## Contribution

It provides a novel explicit construction of diffusions with a prescribed marginal at a random time, connecting martingale diffusions, spectral theory, and coupling methods.

## Key findings

- Explicit construction of diffusions with given marginal at exponential time
- Three different proofs including spectral theory and coupling
- Connections to Skorokhod embedding problem

## Abstract

We solve explicitly the following problem: for a given probability measure mu, we specify a generalised martingale diffusion X which, stopped at an independent exponential time T, is distributed according to mu. The process X is specified via its speed measure m. We present three proofs. First we show how the result can be derived from the solution of Bertoin and Le Jan (1992) to the Skorokhod embedding problem. Secondly, we give a proof exploiting applications of Krein's spectral theory of strings to the study of linear diffusions. Finally, we present a novel direct probabilistic proof based on a coupling argument.