The most visited point of a closed set by Brownian motion
This is an interesting and stimulating paper. \textit{R. F. Bass} and \textit{P. S. Griffin} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 70, 417-436 (1985; Zbl 0554.60076)] have studied the most visited site for a real valued Brownian motion \(B=(B_t:t\geq0)\). The present paper is concerned with the most visited site of a fixed closed set \(F\subseteq \mathbb{R}\) that contains 0. More precisely, let \((L^x_t:x\in\mathbb{R}\), \(t\geq 0)\) be the jointly continuous local times and \(L^F_t=\sup_{x\in F}L^x_t\). Roughly, the point \(A^F_t\) in \(F\) which is the most visited by \(B\) up to time \(t\) is the location where \(L^\cdot_t\) reaches its maximum on \(F\). It is shown that a.s. for every \(t>0\), there is a unique point in \(F\) where this maximum is achieved, except for a set of at most countably many times for which there are exactly two such locations. This yields a càdlàg process \(A^F=(A^F_t:t\geq 0)\) which stays constant over every interval on which \(L^F\) is constant; moreover it is shown that \(A^F\) is a purely discontinuous process with bounded variation. In the special case when \(F=\mathbb{R}\), the author proves that \(L^\mathbb{R}\) coincides with the local time at 0 of the semimartingale \(B-L^\mathbb{R}\) and that the set of jump times of \(A^\mathbb{R}\) has no isolated points.
discontinuous process with bounded variation
scientific article; zbMATH DE number 1018416
scientific article