Time-Homogeneous Diffusions with a Given Marginal at a Random Time

Details Time-Homogeneous Diffusions with a Given Marginal at a Random Time Time-Homogeneous Diffusions with a Given Marginal at a Random Time

2009-12-09

arxiv.org/abs/0912.1719v1

Mathematics

Probability

15 pages, 2 figures

Scientific paper

We solve explicitly the following problem: for a given probability measure mu, we specify a generalised martingale diffusion X which, stopped at an independent exponential time T, is distributed according to mu. The process X is specified via its speed measure m. We present three proofs. First we show how the result can be derived from the solution of Bertoin and Le Jan (1992) 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.

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