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

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 xi (1)(x) of the n-dimensional process xi(x) = {xi (1)(x), aEuro broken vertical bar, xi (n) (x)} first drops under a given level on interval [x, x + Delta x) constraint (xi (2), aEuro broken vertical bar, xi (n) ) a D aS, 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 xi (1)(x) under a given level on interval [x, x + Delta x) this component had already crossed this level a certain number of times.