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

Alexander M.G. Cox , David Hobson , Jan Obłój

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

Alexander M.G. Cox, David Hobson, Jan Obłój. Time-homogeneous diffusions with a given marginal at a random time. ESAIM: Probability and Statistics, 2010, 15, pp.S11-S24. ⟨10.1051/ps/2010021⟩. ⟨hal-00632716⟩