Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60H10 Stochastic ordinary differential equations, See Also {
- 60J65 Brownian motion, See also {58G32}
- 60J60 Diffusion processes, See also {58G32}

**Résumé:** We generalize the notion of brownian bridge. More precisely, we study a
standard brownian motion for which a certain functional is conditioned to
follow a given law. Such processes appear as weak solutions of stochastic
differential equations which we call conditioned stochastic differential
equations. The link with the theory of initial enlargement of filtration is
made and after a general presentation several examples are studied: the
conditioning of a standard brownian motion by its value at a given date, the
conditioning of a geometric brownian motion with negative drift by its
quadratic variation and finally the conditioning of a standard brownian
motion by its first hitting time of a given level. The conditioned
stochastic differential equation associated with the quadratic variation of
the geometric brownian motion allows us to give a new proof of the extension
of the Matsumoto-Yor's $\frac{\langle X\rangle }{X}$ theorem. Moreover, we
show that the set of all the bridges over a given diffusion Z can be
parametrized by a generalized Burger's equation whose solutions are related
by the Hopf-Cole transformation to the positive space-time harmonic
functions of Z. As a consequence of this, we deduce that the set of
diffusions which have the same bridges as Z is parametrized by the positive
eigenfunctions of the generator of Z.

**Mots Clés:** *Brownian bridge ; Stochastic differential equation ; Initial enlargement of
filtrations ; Filtering ; Burger's equation ; Matsumoto-Yor's $\frac{\langle
X\rangle }{X}$ property*

**Date:** 2001-04-04

**Prépublication numéro:** *PMA-649*

**Pdf file :** PMA-649.pdf