International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 35, Pages 2221-2239

Asymptotic solutions of diffusion models for risk reserves

S. Shao

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

Received 22 August 2002

Copyright © 2003 S. Shao. 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 risk reserves which account for the investment income earned and for the inflation experienced on claim amounts. After we defined the process of the conditional probability of ruin over finite time and imposed the appropriate boundary conditions, classical results from the theory of diffusion processes turn the stochastic differential equation to a special class of initial and boundary value problems defined by a linear diffusion equation. Armed with asymptotic analysis and perturbation theory, we obtain the asymptotic solutions of the diffusion models (possibly degenerate) governing the conditional probability of ruin over a finite time in terms of interest rate.