The Cereteli-Davis solution to the \(H^ 1\)-embedding problem and an optimal embedding in Brownian motion. (English) Zbl 0607.60071

Perkins, Edwin

Stochastic processes, 5th Semin., Gainesville/Fla. 1985, Prog. Probab. Stat. 12, 172-223 (1986).

[For the entire collection see Zbl 0591.00011.]
The first half of this paper is an exposition of the following result, due to O. D. Cereteli [Soobshch. Akad. Nauk Gruz. SSR 81, 281-283 (1976; Zbl 0319.42005)] and (independently) B. Davis [Trans. Am. Math. Soc. 261, 211-233 (1980; Zbl 0438.42010)]:
Suppose \(\mu\) is a mean-zero probability on the line; a necessary and sufficient condition for the existence of a stopping time T, and a Brownian motion B, such that \(B_ T\) has law \(\mu\) and \(B^*_ T=\sup \{| B_ s|:\) \(s\leq T\}\) is integrable, is: \[ \int^{\infty}_{0}\lambda^{-1}| \int^{\infty}_{-\infty}x 1_{\{| x| \geq \lambda \}}d\mu (x)| d\lambda <\infty. \] The second half shows how to select T so as to minimize stochastically both \(\sup_{s\leq T}B_ s\) and \(-\inf_{s\leq T}B_ s\).

Reviewer: R.W.R.Darling

MSC:

60J65	Brownian motion
60G46	Martingales and classical analysis

Keywords:

Skorokhod embedding; stopping-time

Citations:

Zbl 0591.00011; Zbl 0319.42005; Zbl 0438.42010