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Stopping times of Bessel processes. (English) Zbl 0645.60082

Let \(X^ x_{\alpha}\) be a Bessel process with parameter \(\alpha\), starting at \(x\geq 0\). L. Gordon [Ann. Math. Stat. 43, 1927-1934 (1972; Zbl 0267.60046)] obtained \(L^ p\) inequalities which relate stopping times to stopping places for the case \(\alpha =1\), \(x=0\) and \(p>\). W. A. Rosenkrantz and S. Sawyer [Z. Wahrscheinlichkeitstheor. Verw. Geb. 41, 145-151 (1977; Zbl 0376.60047)] extended them to \(\alpha >0\), \(x=0\) and \(p\geq 1\). D. L. Burkholder [Adv. Math. 26, 182-205 (1977; Zbl 0372.60112)] obtained results for \(\alpha\) a positive integer, \(x\geq 0\) and \(p>0.\)
Here we consider arbitrary starting points x, \(\alpha >0\) and \(p>0\). The \(L^ p\) inequalities are valid for \(\alpha\geq 2\) with \(p>0\), and also for \(0<\alpha <2\) with \(p>(2-\alpha)/2\). Examples are constructed to show that for \(0<\alpha <2\) with \(p\leq (2-\alpha)/2\), the \(L^ p\) inequalities cannot hold.

MSC:

60J60 Diffusion processes
60G40 Stopping times; optimal stopping problems; gambling theory
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