Some results about boundary crossing for Brownian motion
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.
scientific article; zbMATH DE number 1884308