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 $\boldsymbol\xi(x)=\{\xi_1(x),\ldots,\xi_n(x)\}$ first drops under a given level on interval $[x,x+\triangle x)$ constraint$(\xi_2,\ldots,\xi_n)\in D\subset R^{n-1}$ 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 $[x,x+\triangle x)$ this component had already crossed this level a certain number of times.