Embedding laws in diffusions by functions of time
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Cox, A. M. G. [1] ; Peskir, G. [2]
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[1] University of Bath
University of Bath
Reino Unido
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[2] University of Manchester
University of Manchester
Reino Unido
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 5, 2015, págs. 2481-2510
- Idioma: inglés
- DOI: 10.1214/14-AOP941
- Enlaces
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Resumen
We present a constructive probabilistic proof of the fact that if B=(Bt)t≥0 is standard Brownian motion started at 0, and μ is a given probability measure on R such that μ({0})=0, then there exists a unique left-continuous increasing function b:(0,∞)→R∪{+∞} and a unique left-continuous decreasing function c:(0,∞)→R∪{−∞} such that B stopped at τb,c=inf{t>0|Bt≥b(t) or Bt≤c(t)} has the law μ. The method of proof relies upon weak convergence arguments arising from Helly’s selection theorem and makes use of the Lévy metric which appears to be novel in the context of embedding theorems. We show that τb,c is minimal in the sense of Monroe so that the stopped process Bτb,c=(Bt∧τb,c)t≥0 satisfies natural uniform integrability conditions expressed in terms of μ. We also show that τb,c has the smallest truncated expectation among all stopping times that embed μ into B. The main results extend from standard Brownian motion to all recurrent diffusion processes on the real line.
