Singular Perturbation Analysis of the Differential Equations for the Exit Probability and Exit Time in One Dimension

Abstract

In this chapter we construct asymptotic solutions for one-dimensional exit problems directly from the differential equation using the singular perturbation method. However, in some of these problems the standard method fails and we have to introduce more refined methods such as the WKB-method or a variational approach. Of course we may as well produce an exact solution and make an asymptotic approximation with the Laplace method applied to this exact solution. We choose to carry out a singular perturbation analysis because this method can be employed in higher dimensional problems for which an exact solution is not available. The present class of one-dimensional problems offers an excellent opportunity to develop the appropriate tools and to explore the scope of the method.