Estimating the probability that a multidimensional random process reaches the boundary of a region

Abstract

We consider the problem of finding the probability of the event that a continuous random process reaches the boundary of a region first time on a given range of the independent variable. We propose a new approach to estimating the said probability related to studying the so-called conditional probabilities of a horizontal window: a) conditional probability of the event that at the moment when component ? 1(x) of the n-dimensional process ?(x) = {? 1(x), ?, ? n (x)} first drops under a given level on interval [x, x + Δx) constraint (? 2, ?, ? n ) ? D ? R n?1 holds, where D is a given region, given that it has dropped under this level at all; b) conditional probability, under the same condition as a), of the event that up until the moment of the first entry of component ? 1(x) under a given level on interval [x, x + Δx) this component had already crossed this level a certain number of times.