Path collapse for multidimensional Brownian motion with rebirth
Consider a bounded domain \({\mathcal R}\) in \({\mathbb R}^d\), \(d>1\), which is either a convex set with piecewise smooth boundary or a polyhedron, containing two points \(x\) and \(y\). Let \((z_x (t))_{t\in {\mathbb R}_+}\) and \((z_y (t))_{t\in {\mathbb R}_+}\) be two processess constructed from the same Brownian path starting at \(x\), resp. \(y\), and reborn in \(x\), resp. \(y\), when they hit the boundary of \({\mathcal R}\). In the main result of this paper the authors identify a grid \(G({\mathcal R})\) such that \((z_x (t))_{t\in {\mathbb R}_+}\) and \((z_y (t))_{t\in {\mathbb R}_+}\) meet with positive probability strictly less than one if \(x-y\in G({\mathcal R})\), and probability zero if \(x-y\in {\mathcal R}\setminus G({\mathcal R})\). The grid \(G({\mathcal R})\) is contructed by reunion of countably many \((d-1)\)-dimensional hypersurfaces of positive surface area. This extends a previous one-dimensional result by the same authors [J. Theor. Probab. 16, No. 1, 147--159 (2003; Zbl 1015.60071)] in which they showed that \((z_x (t))_{t\in {\mathbb R}_+}\) and \((z_y (t))_{t\in {\mathbb R}_+}\) meet with probability one or zero, depending on whether \(x-y\) belongs or does not belong to a grid constructed by integer linear combinations of the endpoints of \({\mathcal R}\). The result can be used to study the long-term behaviour of barrier options.
scientific article; zbMATH DE number 2142314