Hobson, David (David G.) (2015) Integrability of solutions of the Skorokhod embedding problem for diffusions. Electronic Journal of Probability, 20 . 83. doi:10.1214/EJP.v20-4121 ISSN 1083-6489.

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Abstract

Suppose X is a time-homogeneous diffusion on an interval IX⊆R and let μ be a probability measure on IX. Then τ is a solution of the Skorokhod embedding problem (SEP) for μ in X if τ is a stopping time and Xτ∼μ. There are well-known conditions which determine whether there exists a solution of the SEP for μ in X. We give necessary and sufficient conditions for there to exist an integrable solution. Further, if there exists a solution ofthe SEP then there exists a minimal solution. We show that every minimal solution of the SEP has the same first moment. When X is Brownian motion, there exists an integrable embedding of μ if and only if μ is centred and in L2. Further,every integrable embedding is minimal. When X is a general time-homogeneous diffusion the situation is more subtle. The case with drift can be reduced to the local martingale case by a change of scale. If Y is a diffusion in natural scale, and if the target law is centred, then as in the Brownian case, there is an integrable embedding if the target law satisfies an integral condition. However, unlike in the Brownian case, there exist integrable embeddings of target laws which are not centred. Further, there exist integrable embeddings which are not minimal. Instead, if there exists an integrable embedding, then the set of minimal embeddings is the set of embeddings such that the mean equals a certain quantity, which we identify.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Statistics
Library of Congress Subject Headings (LCSH):	Embedding theorems, Minimalist theory, Homogeneous spaces, Diffusion processes
Journal or Publication Title:	Electronic Journal of Probability
Publisher:	University of Washington. Dept. of Mathematics
ISSN:	1083-6489
Official Date:	10 August 2015
Dates:	Date Event 10 August 2015 Published
Volume:	20
Number of Pages:	26
Article Number:	83
DOI:	10.1214/EJP.v20-4121
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	31 December 2015
Date of first compliant Open Access:	31 December 2015

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