International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 14, Pages 721-739
doi:10.1155/S016117120430431X
Asymptotic and numerical solutions for diffusion models for compounded risk reserves with dividend payments
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.
Abstract
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.