The First Achievement of a Given Level by a Random Process

Abstract

We propose a scheme for finding the probabilities of events related to crossings of a level by a random process. Using this scheme, we estimate the probability that the first achievement of a given level by the component $y_{1}(x)$ of an n-dimensional continuous process ${\mathbf { y}}(x)\!=\!\{y_{1}(x),\ldots,y_{n}(x)\}$ occurs at some moment $x^{*}$ from a given interval $(x',x'')$ and, at this moment $x^{*}$ , the other components $y_{2}(x^{*}),\ldots,y_{n}(x^{*})$ satisfy given constraints. The need for estimating the above-mentioned probability arises, in particular, in the problems of ensuring the safety of an aircraft landing.