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On the behaviour of certain Bessel functionals. An application to a class of stochastic differential equations. (English) Zbl 0627.60070
Let \((B_ t)\) denote a d-dimensional Bessel process. We investigate the behaviour of integral functionals of the form \(\int^{t}_{0}f(B_ s)ds\), \(t\in [0,\infty]\), where f is a non-negative measurable function. If \(B_ 0=a>0\) then \(\int^{t}_{0}f(B_ s)ds<\infty\), \(0\leq t<\infty\), a.s. if and only if f is locally integrable in (0,\(\infty)\). For \(B_ 0=0\) a sufficient condition for the a.s. convergence of the functionals for \(0\leq t<\infty\) is given. Furthermore, we state conditions for the a.s. divergence of \(\int^{\infty}_{0}f(B_ s)ds\). An application to a class of stochastic differential equations is indicated.
Some of the results are related to an earlier paper [Lect. Notes Contr. Inf. Sci. 36, 47-55 (1981; Zbl 0468.60077)], where we were concerned with analogous functionals of the one-dimensional Wiener process.
For further discussions of the behaviour of integral functionals of the Bessel process see also the interesting paper by Xing-Xiong Xue, A zero-one law for integral functionals of the Bessel process (to appear).

MSC:
60J55 Local time and additive functionals
60J60 Diffusion processes
60G17 Sample path properties
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