International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 14, Pages 721-739

Asymptotic and numerical solutions for diffusion models for compounded risk reserves with dividend payments

S. Shao and C. L. Chang

Department of Mathematics, Cleveland State University, Cleveland 44115, OH, USA

Received 17 April 2003

Copyright © 2004 S. Shao and C. L. Chang. 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.


We study a family of diffusion models for compounded risk reserves which account for the investment income earned and for the inflation experienced on claim amounts. We are interested in the models in which the dividend payments are paid from the risk reserves. After defining the process of conditional probability in finite time, martingale theory turns the nonlinear stochastic differential equation to a special class of boundary value problems defined by a parabolic equation with a nonsmooth coefficient of the convection term. Based on the behavior of the total income flow, asymptotic and numerical methods are used to solve the special class of diffusion equations which govern the conditional ruin probability over finite time.