Calculation of finite time ruin probabilities for some risk models

In this paper we discuss the numerical calculation of finite time ruin probabilities for two particular insurance risk models. The first model allows for the investment at a fixed rate of interest of the surplus whenever this is above a given level. This is related to a model studied by Embrechts and Schmidli [Embrechts, P., Schmidli, H., 1994. Ruin estimation for a general insurance risk model. Adv. Appl. Probability 26 (2), 404-422] and by Schmidli [Schmidli, H., 1994a. Corrected diffusion approximations for a risk process with the possibility of borrowing and investment. Schweizerische Vereinigung der Versicherungsmathematiker. Mitteilungen (1), 71-82; Schmidli, H., 1994b. Diffusion approximations for a risk process with the possibility of borrowing and investment. Commun. Stat. Stochastic Models 10 (2), 365-388]. Our second model is the classical risk model but with the insurer's premium rate depending on the level of the surplus. In our final section, we discuss the extension of the these models to allow for the parameters to change over time in a deterministic way. Our methodology for calculating finite time ruin probabilities is to bound the surplus process by discrete-time Markov chains; the average of the bounds gives an approximation to the ruin probability. This approach was used by the authors in a previous paper, Cardoso and Waters [Cardoso, R.M.R., Waters, H.R., 2003. Recursive calculation of finite time ruin probabilities under interest force. Insurance Math. Econ. 33 (3), 659-676], which considered a risk process with interest earned on the surplus. (c) 2004 Elsevier B.V. All rights reserved.