Penalising Brownian paths. Dedicated to Frank Knight (1933-2007).
type: Item
instance of: scholarly article
title: Penalising Brownian paths. Dedicated to Frank Knight (1933-2007).
zbMATH Open document ID: 1190.60002
DOI: 10.1007/978-3-540-89699-9
author: Bernard Roynette
author: Marc Yor
published in: Lecture Notes in Mathematics
publication date: 2009-01-15T00:00:00Z
review text: Penalising a stochastic process \(X=(X_t,\mathcal F_t)_{t\geq 0}\) means to alter the (infinite dimensional) law \(\mathbb P\) of \(X\) by appropriate weights in such a way that some of the properties of the original process change in a prescribed way. These changes can be quite radical, e.g.\ one can create Brownian-like stochastic processes whose overall maximum is finite or one could condition processes on sets which are of measure zero under \(\mathbb P\). Among the most prominent penalisations are Doob's \(h\)-transform and Feynman-Kac transforms. In all concrete situations, \(X\) is either (\(d\)-dimensional) Brownian motion or a Bessel process. As usual, only the canonical version of these processes are defined on \(\Omega=C([0,\infty),\mathbb R^d)\). The general penalisation set-up uses some \(\mathbb R_+\)-valued process \(\Delta_t, t\geq 0\) on \((\Omega,\mathcal F_\infty)\), \(\mathcal F_\infty = \bigvee_{t\geq 0}\mathcal F_t\), which need not be \(\mathcal F_t\)-adapted and which satisfies \(0<\mathbb E_x\Delta_t < \infty\). To penalise the laws \(\mathbb P_x\), define \[ \mathbb P_x^{(t)}(\cdot) = \frac{\Delta_t}{\mathbb E_x\Delta_t}\cdot \mathbb P_x(\cdot)\quad\text{on}\quad \mathcal F_t,\;t>0, \] and study the (existence of the) limit \(t\to\infty\) in a suitable way. This set-up reminds of Doob's \(h\)-transform, but the procedure is not projective, i.e., it does not define a new probability measure on \(\Omega\) without passing to the limit \(t\to\infty\). Often the process \(\Delta_t\) is of the form \(h(\Gamma_t)\). In this case one writes \(\mathbb P^{(h,t)}_x\); if \(\Gamma_t\) is real-valued and increasing, it is often possible to show that \(\mathbb P^{(h,t)}\underset{t\to\infty}\rightarrow\mathbb Q^{(h)}\) exists and defines a new probability measure \(\mathbb Q^{(h)}\) on \((\Omega,\mathcal F_\infty)\). Clearly, \(\mathbb Q^{(h)}(\Gamma_\infty\in dy) = h(y)\,dy\). Examples of \(\Gamma_t\) include (in one dimension) the supremum process \(\su…
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60G30
zbMATH DE Number: 5492707
zbMATH Keywords: Brownian motion
zbMATH Keywords: Wiener measure
zbMATH Keywords: penalization
zbMATH Keywords: conditioning
zbMATH Keywords: statistical mechanics
zbMATH Keywords: Feynman-Kac formula
zbMATH Keywords: Sturm-Liouville equation
zbMATH Keywords: Bessel process
zbMATH Keywords: limit theorems
reviewed by: R. L. Schilling
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-540-89699-9
OpenAlex ID: W12368450
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description: scientific article