### Abstract

We solve explicitly the following problem: for a given probability measure μ, we specify a generalised martingale diffusion (X_t) which, stopped at an independent exponential time T, is distributed according to μ. The process (X_t) is specified via its speed measure m. We present two heuristic arguments and three proofs. First we show how the result can be derived from the solution of [Bertoin and Le Jan, Ann. Probab. 20 (1992) 538–548.] 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.

Original language | English |
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Pages (from-to) | S11-S24 |

Journal | ESAIM: Probability and Statistics |

Volume | 15 |

DOIs | |

Publication status | Published - Feb 2011 |

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## Cite this

Cox, A. M. G., Hobson, D., & Obłój, J. (2011). Time-homogeneous diffusions with a given marginal at a random time.

*ESAIM: Probability and Statistics*,*15*, S11-S24. https://doi.org/10.1051/ps/2010021