Volume 15, 2011
Supplement: In honor of Marc Yor
|Page(s)||S11 - S24|
|Published online||19 May 2011|
Time-homogeneous diffusions with a given marginal at a random time
Dept. of Mathematical Sciences,
University of Bath, Bath, BA2 7AY, UK
A.M.G.Cox@bath.ac.uk; web: www.maths.bath.ac.uk/~mapamgc/
2 Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK
D.Hobson@warwick.ac.uk; web: www.warwick.ac.uk/go/dhobson/
3 Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK
email@example.com; web: www.maths.ox.ac.uk/~obloj/
Revised: 25 February 2010
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.
Mathematics Subject Classification: 60G40 / 60J60
Key words: Time-homogeneous diffusion / generalised diffusion / exponential time / Skorokhod embedding problem / Bertoin-Le Jan stopping time
© EDP Sciences, SMAI, 2011
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