Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II
Authors: Roynette Bernard , Vallois Pierre , Yor Marc 
Published: August 2006

Let us denote by \documentclass{aastex}
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\begin{document}
\(P_0\)
\end{document} the Wiener measure defined on the canonical space \documentclass{aastex}
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\begin{document}
\(\big(\Omega=\mathcal{C}(\mathbb{R}_+, \mathbb{R})\)
\end{document}, \documentclass{aastex}
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\begin{document}
\({(X_t)}_{t \ge 0}\)
\end{document}, \documentclass{aastex}
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\begin{document}
\({(\mathcal{F}_t)}_{t \ge 0}\big)\)
\end{document}, and let \documentclass{aastex}
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\begin{document}
\((S_t)\)
\end{document} (resp. \documentclass{aastex}
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\begin{document}
\((I_t)\)
\end{document}), be the one sided-maximum (resp. minimum), \documentclass{aastex}
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\begin{document}
\((L^0_t)\)
\end{document} the local time at \documentclass{aastex}
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\begin{document}
\(0\)
\end{document}, and \documentclass{aastex}
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\begin{document}
\((D_t)\)
\end{document} the number of down-crossings from \documentclass{aastex}
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\begin{document}
\(b\)
\end{document} to \documentclass{aastex}
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\begin{document}
\(a\)
\end{document} (with \documentclass{aastex}
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\begin{document}
\(b≯a\)
\end{document}). Let \documentclass{aastex}
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\begin{document}
\(f\: \mathbb{R} \times \mathbb{R}^d \to \left]0, +\infty\right[\)
\end{document} be a Borel function, and \documentclass{aastex}
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\begin{document}
\((A_t)\)
\end{document} be a process chosen within the set \documentclass{aastex}
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\begin{document}
\(\big\{(S_t)\)
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\((S_t,t)\)
\end{document}; \documentclass{aastex}
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\begin{document}
\((L^0_t)\)
\end{document}; \documentclass{aastex}
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\begin{document}
\((S_t,I_t,L^0_t)\)
\end{document}; \documentclass{aastex}
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\begin{document}
\((D_t)\big\}\)
\end{document}, which consists of 5 elements. We prove a penalization result: under suitable assumptions on \documentclass{aastex}
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\begin{document}
\(f\)
\end{document}, there exists a positive \documentclass{aastex}
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\begin{document}
\(\big((\mathcal{F}_t),P_0\big)\)
\end{document}-martingale \documentclass{aastex}
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\begin{document}
\((M_t^{f})\)
\end{document}, starting at \documentclass{aastex}
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\begin{document}
\(1\)
\end{document}, such that \begin{equation} \label{ab1} \lim_{t\to \infty}\frac{E_0 \big[1_{\Gamma_s}f(X_t,A_t)\big]}{E_0\big[f(X_t,A_t)\big]}=Q_0^{f}(\Gamma_s):= E_0 [1_{\Gamma_s} M_s^{f}],\quad \forall \Gamma_s \in \mathcal{F}_s\ \text{and}\ s\ge 0. \end{equation} We determine the law of \documentclass{aastex}
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\begin{document}
\((X_t)\)
\end{document} under the p.m. \documentclass{aastex}
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\(Q_0^{f}\)
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\begin{document}
\((\Omega, \mathcal{F}_\infty)\)
\end{document} by \eqref{ab1}. For the \documentclass{aastex}
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\(1^{\rm st}, 3^{\rm rd}\)
\end{document} and \documentclass{aastex}
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\(5^{\rm th}\)
\end{document} elements of the set, we prove first that \documentclass{aastex}
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\(Q_0^{f} (A_{\infty} < \infty)=1\)
\end{document}, and more generally \documentclass{aastex}
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\begin{document}
\(Q_0^{f} (0<g < \infty)=1\)
\end{document} where \documentclass{aastex}
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\(g= \sup\, \{s≯0\)
\end{document}, \documentclass{aastex}
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\(A_s=A_{\infty}\}\)
\end{document} \vspace{2pt} (with the convention \documentclass{aastex}
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\(\sup \emptyset=0\)
\end{document}). Secondly, we split the trajectory of \documentclass{aastex}
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\((X_t)\)
\end{document} into \vspace{-2pt} two parts: \documentclass{aastex}
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\({(X_t)}_{0\le t \le g}\)
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\({(X_{t+g})}_{ t \ge 0}\)
\end{document}, and we describe their laws under \documentclass{aastex}
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\(Q_0^{f}\)
\end{document}, conditionally on \documentclass{aastex}
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\begin{document}
\(A_{\infty}\)
\end{document}. For the \documentclass{aastex}
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\(2^{\rm nd}\)
\end{document} and \documentclass{aastex}
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\(4^{\rm th}\)
\end{document} \vspace{2pt} elements, a similar result holds replacing \documentclass{aastex}
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\begin{document}
\(A_\infty\)
\end{document} by resp. \documentclass{aastex}
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\(S_\infty\)
\end{document} and \documentclass{aastex}
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\begin{document}
\(S_\infty \vee I_\infty\)
\end{document}.