The First Achievement of a Given Level by a Random Process

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$y_{1}(x)$ </tex-math></inline-formula> of an n-dimensional continuous process <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbf { y}}(x)\!=\!\{y_{1}(x),\ldots,y_{n}(x)\}$ </tex-math></inline-formula> occurs at some moment <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{*}$ </tex-math></inline-formula> from a given interval <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(x',x'')$ </tex-math></inline-formula> and, at this moment <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{*}$ </tex-math></inline-formula>, the other components <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$y_{2}(x^{*}),\ldots,y_{n}(x^{*})$ </tex-math></inline-formula> 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.