Boundary crossing probabilities for high-dimensional Brownian motion

Abstract

The two-sided nonlinear boundary crossing probabilities for one-dimensional Brownian motion and related processes have been studied in Fu and Wu (2010) based on the finite Markov chain imbedding technique. It provides an efficient numerical method to computing the boundary crossing probabilities. In this paper we extend the above results for high-dimensional Brownian motion. In particular, we obtain the rate of convergence for high-dimensional boundary crossing probabilities. Numerical results are also provided to illustrate our results.

Reliability analyses of linear two-dimensional consecutive k-type systems

“…Similarly, with n = 5, the unreliability of a linear 2-connected-(2,3)-or-(3,2)-out-of-(3, 5) : F system without overlapping can be given as R(5) = πA 5 ū = 7q 12 − 8q 14 + 2q 15 , which coincides with the identity (by definition) that R(5) = P(A 1 ∪ • • • ∪ A 7 ) with the events A 1 , . .…”

“…FMCIA methods are actually improved versions of recursive methods, as they provide unified explicit forms for different numbers of components, and need less memory and computational time for reliability evaluation. FMCIA also has more diverse applications than just in the study of redundant systems [15, 16]. Chang and Huang [8] investigated the reliability of a linear/circular k -within-consecutive-( r , s )-out-of- system using FMCIA, while Zhao et al [35] similarly studied the reliability of a linear connected-( r , s )-out-of- system.…”

Piecewise linear boundary crossing probabilities, barrier options, and variable annuities

“…Among various probability topics, the crossing probability (or equivalently, non‐crossing probability) of the underlying process for a given boundary has drawn attention from academics and practitioners (see e.g., Fu & Wu, 2010, 2016; Novikov et al, 1999; Siegmund, 1986; Wang & Pötzelberger, 1997, 2007). In particular, considering the piecewise linear boundary crossing probability is not new and has a long history in the literature.…”