The Azéma-Yor embedding in non-singular diffusions.
Let \((X_ {t})\) be a real-valued diffusion process, \(X_ 0=0\), and \(\nu \) a probability measure on \(\mathbb R\). The problem whether there exists a stopping time \(\tau \) for \((X_ {t})\) such that the law of \(X_ \tau \) equals to \(\nu \) has been paid a considerable attention recently. A. V. Skorokhod's famous result concerns the case of a Wiener process \((X_ {t})\); in the paper under review, general non-singular (not necessarily recurrent) diffusions are considered. Let \(\mu ,\sigma :\mathbb R\to \mathbb R\) be Borel functions such that \(\sigma >0\) and both \(\sigma ^ {-2}\) and \(| \mu | \sigma ^ {-2}\) are locally integrable. Let \((X_ {t})\) be the unique weak solution up to an explosion time \(\zeta \) to the stochastic differential equation \(dX = \mu (X)\,dt + \sigma (X)\, dW\), \(X_ 0 = 0\), where \(W\) is a standard Wiener process. Define the scale function \(S\) by \(S(x) = \int ^ {x}_ {0}\exp ( -2\int ^ {u}_ {0}\mu (r)/\sigma ^ 2(r)\,dr)\,du\), \(x \in \mathbb R\). Let the probability measure \(\nu \) satisfy \(\int _ {\mathbb R} | S(x)| \,d\nu (x) <\infty \), set \(m=\int _ {\mathbb R} S(x)\,d\nu (x)\). The authors prove that there exists a stopping time \(\tau \) such that the law of \(X_ \tau \) coincides with \(\nu \) if and only if any one of the following conditions holds: (i) \(S(-\infty )=-\infty \) and \(S(\infty )=\infty \), (ii) \(S(-\infty )=-\infty \), \(S(\infty )<\infty \) and \(m\geq 0\), (iii) \(S(-\infty )>-\infty \), \(S(\infty )=\infty \) and \(m\leq 0\), (iv) \(S(-\infty )>-\infty \), \(S(\infty )<\infty \) and \(m = 0\).
scientific article; zbMATH DE number 2138915
scientific article