Ruin Models Featuring Interest and Diffusion.

摘要

This thesis brings together a number of extensions to risk theory models. Initially, we consider a multi-threshold compound Poisson surplus process with interest earned at a constant rate. More precisely, when the initial surplus is between any two consecutive thresholds, the insurer has the option to choose the respective premium rate and interest rate. Another feature is consideration of the "absolute ruin" scenario, in which one is allowed to pay interest on the current amount of deficit whenever the surplus falls below zero. Starting from the integro-differential equations satisfied by the Gerber-Shiu function that appear in Yang et al. (26), we consider exponentially and phase-type(2) distributed claim sizes, in which cases we are able to transform the integra-differential equations into ordinary differential equations. Subsequently, explicit solutions in series form are provided.;Also, we study the absolute ruin and the standard ruin problems in a Markovian Arrival Process (MAP) risk process with an underlying Continuous Time Markov Chain (CTMC) with m states and phase-type claim amounts. Under a certain restriction, we find the Laplace transform of the Gerber-Shiu function in the absolute ruin model. Moreover, we find the Gerber-Shiu function in some particular cases with two states, where the MAP represents Erlang Inter Arrival Times (Erlang IATs), a Markov Modulated Poisson process (i.e. between two environmental states) and a contagion model.;We also consider a multi-layer compound Poisson surplus process perturbed by diffusion and examine the behavior of the Gerber-Shiu discounted penalty function. We derive the general solution to a certain second order integra-differential equation. This permits us to provide explicit expressions for the Gerber- Shiu function depending on the current surplus level. If the diffusion term converges to zero, the above-mentioned explicit expressions converge weakly to those under the classical Compound Poisson model, provided that the same initial conditions apply.