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Time-homogeneous diffusions with a given marginal at a random time

Published online by Cambridge University Press:  15 October 2010

Alexander M.G. Cox ,
David Hobson  and
Jan Obłój
Alexander M.G. Cox
Affiliation:
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/
David Hobson
Affiliation:
Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK; D.Hobson@warwick.ac.uk; web: www.warwick.ac.uk/go/dhobson/
Jan Obłój
Affiliation:
Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK; obloj@maths.ox.ac.uk; web: www.maths.ox.ac.uk/~obloj/
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Abstract

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.

Keywords

Time-homogeneous diffusiongeneralised diffusionexponential timeSkorokhod embedding problemBertoin-Le Jan stopping time
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. S11 - S24
Copyright
© EDP Sciences, SMAI, 2011

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