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 -