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

¹ 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
obloj@maths.ox.ac.uk; web: www.maths.ox.ac.uk/~obloj/

Received: 27 November 2009
Revised: 25 February 2010

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.