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September 2015 Embedding laws in diffusions by functions of time
A. M. G. Cox, G. Peskir
Ann. Probab. 43(5): 2481-2510 (September 2015). DOI: 10.1214/14-AOP941

Abstract

We present a constructive probabilistic proof of the fact that if $B=(B_{t})_{t\ge0}$ is standard Brownian motion started at $0$, and $\mu$ is a given probability measure on $\mathbb{R}$ such that $\mu(\{0\})=0$, then there exists a unique left-continuous increasing function $b:(0,\infty)\rightarrow\mathbb{R}\cup\{+\infty\}$ and a unique left-continuous decreasing function $c:(0,\infty)\rightarrow\mathbb{R}\cup\{-\infty\}$ such that $B$ stopped at $\tau_{b,c}=\inf\{t>0\vert B_{t}\ge b(t)\mbox{ or }B_{t}\le c(t)\}$ has the law $\mu$. The method of proof relies upon weak convergence arguments arising from Helly’s selection theorem and makes use of the Lévy 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 $B^{\tau_{b,c}}=(B_{t\wedge\tau_{b,c}})_{t\ge0}$ 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.

Citation

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A. M. G. Cox. G. Peskir. "Embedding laws in diffusions by functions of time." Ann. Probab. 43 (5) 2481 - 2510, September 2015. https://doi.org/10.1214/14-AOP941

Information

Received: 1 January 2013; Revised: 1 May 2014; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1335.60150
MathSciNet: MR3395467
Digital Object Identifier: 10.1214/14-AOP941

Subjects:
Primary: 60G40, 60J65
Secondary: 60F05, 60J60

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 5 • September 2015
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