Double boundary crossing result for the brownian motion

Abstract

Abstract In a recent paper a general bounded crossing result for the Brownian motion is obtained for a linear upper boundary based on the method of Kac (1951). Based on his main results we are able to develop in the present paper some simple expressions for crossing probabilities in case of a lower and an upper linear boundary. We will consider a Brownian motion process for the surplus of an insurance portfolio. This surplus must stay between two given bounds. If the surplus will cross the upper boundary, we can pay a dividend. The lower boundary can reflect the influence of control authorities and regulation measurements.