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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 675202, 9 pages
http://dx.doi.org/10.1155/2013/675202
Research Article

The Exit Time and the Dividend Value Function for One-Dimensional Diffusion Processes

1School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China
2China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China

Received 30 August 2013; Accepted 27 October 2013

Academic Editor: Yong Zhou

Copyright © 2013 Peng Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the exit times from an interval for a general one-dimensional time-homogeneous diffusion process and their applications to the dividend problem in risk theory. Specifically, we first use Dynkin’s formula to derive the ordinary differential equations satisfied by the Laplace transform of the exit times. Then, as some examples, we solve the closed-form expression of the Laplace transform of the exit times for several popular diffusions, which are commonly used in modelling of finance and insurance market. Most interestingly, as the applications of the exit times, we create the connect between the dividend value function and the Laplace transform of the exit times. Both the barrier and threshold dividend value function are clearly expressed in terms of the Laplace transform of the exit times.