Some results about boundary crossing for Brownian motion
type: Item
instance of: scholarly article
title: Some results about boundary crossing for Brownian motion
zbMATH Open document ID: 1009.60069
published in: Ricerche di Matematica
publication date: 2003-03-18T00:00:00Z
review text: Let \(\{B(t)\), \(t\geq 0 \}\) be a standard Brownian motion (BM) started at 0, \(S(t)\), \(t\geq 0 \), be a smooth non-random function and let \(\tau_S(x)\) be the first-passage time of \(B(t)\) over \(S(t)\). Denote by \(F_S(t)\) the distribution function of \(\tau_S\) and by \(r_S(t)\) its hazard rate, so that \(F_S(t)=1-\exp\int_0^t r_S(u) du\). The author extends some results of \textit{T. H. Scheike} [J. Appl. Probab. 29, No. 2, 448-453 (1992; Zbl 0806.60065)] about the first-passage time of Brownian motion through a piecewise linear boundary. These results are in order to improve the hazard-rate estimation of the first-passage time distribution of BM over a curved boundary \(S\) by considering a polygonal approximation. Such a procedure works rather satisfactory for concave and convex boundaries. In the case in which analytical results are known the comparison was carried out and the proposed estimates turn out to be closer to the exact values than the estimates obtained by the hazard rate tangent approximation [\textit{G. O. Roberts} and \textit{C. F. Shortland}, Ann. Appl. Probab. 5, No. 2, 446-460 (1995; Zbl 0836.60087)]. The hazard-rate approach is also extended to some other classes of diffusion processes, for instance, certain results are obtained for \(Y(t)+ B(t)\) and jump-diffusion process \(X(t)\) satisfying \(DX(t)=b(X(t)) d(t)+\sigma(X(t)) dB(t)+\varepsilon dN_t\), where \(X(0)=0\), \(\varepsilon>0\), \(N_t\) is a homogeneous Poisson process. Finally, some results concerning the last-passage time over \(S(t)\) are presented.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 1884308
zbMATH Keywords: Brownian motion
zbMATH Keywords: boundary-crossing probability
zbMATH Keywords: first-passage-time
zbMATH Keywords: diffusion process
zbMATH Keywords: hazard-rate approximation
zbMATH Keywords: piecewise-linear boundary
author: Mario Abundo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1884308