Embedding in Brownian motion with drift and the Azéma-Yor construction
Let \(B\) be a standard one-dimensional Brownian motion. According to the classical Skorokhod embedding theorem, for any probability measure \(\mu \) on \(\mathbb R\) with mean zero satisfying \(\sigma ^2=\int x^2 d\mu (x) <\infty \) there exists a stopping time \(T\) such that \(B_{T}\) has the law \(\mu \) and \(\mathbf ET=\sigma ^2\). \textit{J. Azéma} and \textit{M. Yor} [in: Séminaire de probabilités XIII. Lect. Notes Math. 721, 90-115 (1979; Zbl 0414.60055)] proposed a new proof of this result and extended it to recurrent diffusions starting at zero. In the paper under review, a transient diffusion \(X_{t} = \kappa t + B_{t}\), \(\kappa >0\), is considered and the Azéma-Yor method is modified to find for any probability measure \(\mu \) on \(\mathbb R\) satisfying \(\int \exp (-2\kappa x) d\mu (x) =1\) a stopping time \(T\) such that \(X_{T}\) has distribution \(\mu \). Moreover, all measures on \(\left ]-\infty ,\infty \right ]\) that may appear as a law of \(X_{T}\) for a (not necessarily finite) stopping time \(T\) are characterized.
scientific article; zbMATH DE number 1792502
scientific article