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A number of limit laws, which are obtained from various penalisations of the Wiener measure on C (ℝ+,ℝd), are shown to exist, and are described thoroughly, with the help of path decompositions
As an Introduction to this monograph, we present, in a Brownian and Bessel framework, the general problem of penalisation which will be discussed throughout this volume.We sketch a number of examples, the study of which constitutes the different Chapters of the monograph. Finally, we make a list of the martingales which occur as Radon-Nikodym densities between the Wiener measure and the penalised...
In the preceding chapters, we have shown that, in a large class of frameworks, certain families $$ \{ P^{(t)} ,t \ge 0\} $$ of probabilities defined on a filtered space $$ \{ \Omega ,(F_s )_{S \ge 0} \} $$ converge, as t → ∞, to a probability Q, at least in the sense that : $$\forall \;s\; > 0,\;\forall \Lambda _s \in \;F_s ,\;P^{(t)} (\Lambda _S )\mathop \to \limits_{t \to \infty...
Among the various examples of penalisations of Wiener measure discussed in this Monograph, the ones which are obtained by putting a Feynman-Kac type weight with respect to Wiener measure, up to time t, are undoubtedly quite natural, and such transforms of Wiener measure have a long history. In this Chapter, we show that the asymptotic behavior of all these penalised measures may be expressed...
For any integer n = 1, 2, ⋯, limiting laws, as t→∞, for a Bessel process with dimension d (0 < d < 2) penalised by the nth-ranked length of its excursions up to t, or up to the last zero before t, or again up to the first zero after t, are shown to exist, and are characterized. Under these limiting laws Q(n), the canonical process admits a last zero g, and the sequence of the...
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