The tangent approximation

Abstract

In this section we study the tangent approximation in general and derive it for a broad class of boundaries. It will be shown that the tangent approximation holds uniformly over intervals if the boundaries recede to infinity. Therefore by integrating out the densities, approximations for the first exit probabilities can be derived. The sets on which the tangent approximation holds can be finite intervals or the whole real line. This depends on the fact, whether the boundaries belong to the upper or lower class at infinity. All the results about the tangent approximation hold uniformly over all drift directions. A refinement of the tangent approximation, a second order approximation due to Jennen is also given and the quality of the approximation is discussed for some examples.