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.
Mots clés : time-homogeneous diffusion, generalised diffusion, exponential time, Skorokhod embedding problem, Bertoin-Le Jan stopping time
@article{PS_2011__15__S11_0,
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://www.numdam.org/articles/10.1051/ps/2010021/}
}
TY - JOUR AU - Cox, Alexander M. G. AU - Hobson, David AU - Obłój, Jan TI - Time-homogeneous diffusions with a given marginal at a random time JO - ESAIM: Probability and Statistics PY - 2011 SP - S11 EP - S24 VL - 15 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2010021/ DO - 10.1051/ps/2010021 LA - en ID - PS_2011__15__S11_0 ER -
%0 Journal Article %A Cox, Alexander M. G. %A Hobson, David %A Obłój, Jan %T Time-homogeneous diffusions with a given marginal at a random time %J ESAIM: Probability and Statistics %D 2011 %P S11-S24 %V 15 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2010021/ %R 10.1051/ps/2010021 %G en %F PS_2011__15__S11_0
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. http://www.numdam.org/articles/10.1051/ps/2010021/
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