Abstract
We present a constructive probabilistic proof of the fact that if B=(B_t)_{t \ge 0} is standard Brownian motion started at 0 and mu is a given probability measure on R such that mu({0})=0 then there exists a unique leftcontinuous increasing function b and a unique leftcontinuous decreasing function c such that B stopped at tau_{b,c}=inf{t>0 : B_t \ge b(t) or B_t \le c(t)} has the law mu. The method of proof relies upon weak convergence arguments arising from Helly's selection theorem and makes use of the L\'evy metric which appears to be novel in the context of embedding theorems. We show that tau_{b,c} is minimal in the sense of Monroe so that the stopped process satisfies natural uniform integrability conditions expressed in terms of mu. We also show that tau_{b,c} has the smallest truncated expectation among all stopping times that embed mu into B. The main results extend from standard Brownian motion to all recurrent diffusion processes on the real line.
Original language  English 

Pages (fromto)  24812510 
Number of pages  30 
Journal  Annals of Probability 
Volume  43 
Issue number  5 
Early online date  9 Sept 2015 
DOIs  
Publication status  Published  30 Sept 2015 
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Alex Cox
 Department of Mathematical Sciences  Deputy Head of Department
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Probability Laboratory at Bath
Person: Research & Teaching