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From minimal embeddings to minimal diffusions. (English) Zbl 1312.60093

Summary: We show that there is a one-to-one correspondence between diffusions and the solutions of the Skorokhod embedding problem due to J. Bertoin and Y. Le Jan [Ann. Probab. 20, No. 1, 538–548 (1992; Zbl 0749.60038)]. In particular, the minimal embedding corresponds to a “minimal local martingale diffusion”, which is a notion we introduce in this article. Minimality is closely related to the martingale property. A diffusion is minimal if it minimises the expected local time at every point among all diffusions with a given distribution at an exponential time. Our approach makes explicit the connection between the boundary behaviour, the martingale property and the local time characteristics of time-homogeneous diffusions.

MSC:

60J60 Diffusion processes
60J55 Local time and additive functionals
60G44 Martingales with continuous parameter

Citations:

Zbl 0749.60038
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