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Penalizations and some extensions of Pitman’s theorem relative to Brownian motion and its one-sided maximum. (Pénalisations et quelques extensions du théorème de Pitman, relatives au mouvement Brownien et à son maximum unilatère.) (French) Zbl 1124.60034

Émery, Michel (ed.) et al., In memoriam Paul-André Meyer. Séminaire de probabilités XXXIX. Berlin: Springer (ISBN 3-540-30994-2/pbk). Lecture Notes in Mathematics 1874, 305-336 (2006).
Summary: On the canonical Wiener space \(\Omega={\mathcal C}([0,\infty[\to \mathbb{R})\), where \((X_t,t\geq 0)\) denotes the coordinate process, \({\mathcal F}_t=\sigma(X_s,s\leq t)\) its natural filtration and \((W_x,x\in\mathbb{R})\) the family of Wiener measures, we consider several kinds of adapted, \(\mathbb{R}_+\)-valued, integrable functionals \((\Gamma_t,t\geq 0)\), for which we show that, for every fixed \(s>0\) and \(\Lambda_s\in{\mathcal F}_s\), the quantity \(E_x(1_{\Lambda_s}\Gamma_t)/ E_x (\Gamma_t)\) admits a limit, as \(t\to\infty\), and that this limit takes the form \(E_x (1_{\Lambda_s}M_s^x)\) where \((M_s^x,s\geq 0)\) is an \(({\mathcal F}_s,s\geq 0,W_x)\) martingale. This allows us to define, on \((\Omega,{\mathcal F}_\infty)\), a probability \(Q_x\) via the formula: \[ Q_x(\Lambda_s):= E_x(1_{\Lambda_s}M_s^x) \quad(\Lambda_s\in{\mathcal F}_s). \] We then describe precisely the process \((X_t,t\geq 0)\) under the probability \(Q_x\). In general, this process is not Markovian on its own filtration, but several path decompositions and Markov properties are obtained.
For the entire collection see [Zbl 1092.60003].

MSC:

60G07 General theory of stochastic processes
60J65 Brownian motion
60G44 Martingales with continuous parameter
60H20 Stochastic integral equations
60G17 Sample path properties
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