Geodesic
Time-homogeneous diffusions with a given marginal at a random time
ESAIM: Probability and Statistics, Tome 15 (2011), pp. S11-S24

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We solve explicitly the following problem: for a given probability measure μ, we specify a generalised martingale diffusion (Xt) which, stopped at an independent exponential time T, is distributed according to μ. The process (Xt) 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.

DOI : 10.1051/ps/2010021
Classification : 60G40, 60J60
Keywords: time-homogeneous diffusion, generalised diffusion, exponential time, Skorokhod embedding problem, Bertoin-Le Jan stopping time 
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     author = {Cox, Alexander M. G. and Hobson, David and Ob{\l}\'oj, Jan},
     title = {Time-homogeneous diffusions with a given marginal at a random time},
     journal = {ESAIM: Probability and Statistics},
     pages = {S11--S24},
     publisher = {EDP-Sciences},
     volume = {15},
     year = {2011},
     doi = {10.1051/ps/2010021},
     mrnumber = {2817342},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2010021/}
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Cox, Alexander M. G.; Hobson, David; Obłój, Jan. Time-homogeneous diffusions with a given marginal at a random time. ESAIM: Probability and Statistics, Tome 15 (2011), pp. S11-S24. doi: 10.1051/ps/2010021

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