TY - JOUR

T1 - Time-homogeneous diffusions with a given marginal at a random time

AU - Cox, Alexander M G

AU - Hobson, David

AU - Obłój, Jan

PY - 2011/2

Y1 - 2011/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84855678731&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1051/ps/2010021

U2 - 10.1051/ps/2010021

DO - 10.1051/ps/2010021

M3 - Article

SN - 1292-8100

VL - 15

SP - S11-S24

JO - ESAIM: Probability and Statistics

JF - ESAIM: Probability and Statistics

ER -