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On a transformation of Brownian motion by Jeulin and Yor. (Sur une transformation du mouvement brownien due à Jeulin et Yor.) (French) Zbl 0811.60064

Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXVIII. Berlin: Springer-Verlag. Lect. Notes Math. 1583, 98-101 (1994).
Let \((X_ t\); \(0\leq t\leq 1)\) be the unique strong solution in \(\mathbb{R}^ d\) of \[ X_ t^ i= X_ 0^ i+ \sum_ \alpha \int_ 0^ t a_ \alpha^ i (s,X_ s) dB_ s^ \alpha+ \int_ 0^ t b^ i(s, X_ s)ds, \qquad 0\leq t\leq 1, \] where \(1\leq i\leq d\), \((a_ \alpha^ i)_{1\leq\alpha \leq l}\) and \(b^ i\) are smooth. Let \(p_ t\) be the density of \(X_ t\). The author proves that the process \(\widehat {B}\) defined by: \[ \widehat {B}_ t^ \alpha= B_{1-t}^ \alpha- B_ 1^ \alpha- \int_ 0^ t {\textstyle {{\text{div}(pa_ \alpha)} \over p}} (1-s, X_{1-s})ds, \qquad \alpha=1,2,\dots, l, \] is an \(R^ l\)-valued Brownian motion with respect to \(\widehat {\mathcal F}= (\widehat {\mathcal F}_ t\); \(t\geq 0)\), where \(\widehat {\mathcal F}_ t= \sigma(X_{1-t}, (B_ s-B_ 1\); \(1-t\leq s\leq 1))\). Moreover the author proves that \(\widehat {X}_ t= X_{1-t}\) solves \[ \widehat {X}_ t^ i= \widehat {X}_ 0^ i+ \sum_ \alpha \int_ 0^ t a_ \alpha^ i (s,\widehat {X}_ s) d\widehat {B}_ s^ \alpha+ \int_ 0^ t \widehat {b}^ i (s,\widehat {X}_ s)ds, \qquad 0\leq t\leq 1, \] \(\widehat {b}^ i\) being explicit.
For the entire collection see [Zbl 0797.00020].
Reviewer: P.Vallois (Nancy)

MSC:

60J65 Brownian motion
60G44 Martingales with continuous parameter
60J25 Continuous-time Markov processes on general state spaces
60J60 Diffusion processes
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