An Asymptotic Expression for the Probability of Ruin within Finite Time

Abstract

We consider quantities such as the probability that a two-dimensional random walk crosses the ordinate $y$ for the first time to the left of the abscissa $x$, and describe the asymptotic behaviour as $x$ and $y$ tend to $\infty$. The result is applied to the risk reserve process of insurance mathematics as well as to one-dimensional random walks.