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Study of a functional linked to Bessel bridge. (Étude d’une fonctionnelle liée au pont de Bessel.) (French) Zbl 0842.60076
The authors investigate the behaviour of $$I_t (x,y):= E[\exp \int^t_0 l( X^{x,y}_{s,t} )ds]$$, where (i) $$(X^{x,y}_{s,t}$$; $$0\leq s\leq t)$$ denotes the $$n$$-dimensional Bessel bridge from $$x$$ to $$y$$ over time $$t$$, $$n> 2$$, (ii) $$l: \mathbb{R}_+\to \mathbb{R}_+$$ is a bounded function verifying: $$\gamma_-< l(x)< \gamma_+$$ (resp. $$c_1\leq (l(x)- {{k(p)} \over {x^2}}) x^{2+ q-p}\leq c_2)$$ for any $$x\in [0,1 ]$$ (resp. $$[1, +\infty[)$$, where $$\gamma_-$$, $$\gamma_+$$, $$c_1$$ and $$c_2$$ are explicit and fixed constants, $$k(p)= p(n-p- 2)/2$$, $$0<p< (n-2)/ 2$$, $$q>p$$. The authors show that for any $$y_0>0$$, there exists a positive constant $$C$$ (depending only on $$l$$, $$y_0$$, $$n$$, $$p)$$ such that $$\forall t>0$$, $$\forall x\geq 0$$, $$0\leq y\leq y_0$$, $C^{-1} \Biggl( {{1+x+t} \over {1+x}} \Biggr)^p\leq I_t (x,y)\leq C \Biggl( {{1+x+t} \over {1+x}} \Biggr)^p.$ This result allows the authors to give explicit lower and upper bounds of the density of hyperbolic diffusions on $$\mathbb{R}_+$$, i.e. the generator is the differential operator $H_{\alpha, \beta} (f) (r)= (f'' (r)+ \gamma_{\alpha, \beta} (r) f' (r))/2, \qquad \gamma_{\alpha, \beta} (r)= \alpha \coth r+ 2\beta \coth 2r.$
Reviewer: P.Vallois (Nancy)

MSC:
 60J60 Diffusion processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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