Modern Stochastics: Theory and Applications

Given a compound mixed renewal process S under a probability measure P, we provide a characterization of all progressively equivalent martingale probability measures Q on the domain of P, that convert S into a compound mixed Poisson process. This result extends earlier works of Delbaen and Haezendonck, Lyberopoulos and Macheras, and the authors, and enables us to find a wide class of price processes satisfying the condition of no free lunch with vanishing risk. Implications to the ruin problem and to the computation of premium calculation principles in an arbitrage-free insurance market are also discussed.

Based on a discrete version of the Pollaczeck–Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber–Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.

The discrete time risk model with two seasons and dependent claims is considered. An algorithm is created for computing the values of the ultimate ruin probability. Theoretical results are illustrated with numerical examples.

We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal risk model. We consider the model with independent, but not necessarily identically distributed, claim sizes and the interoccurrence times. In order to prove the main theorem, we first formulate and prove an auxiliary lemma on large values of a sum of random variables asymptotically drifted in the negative direction.

We deal with a generalization of the classical risk model when an insurance company gets additional funds whenever a claim arrives and consider some practical approaches to the estimation of the ruin probability. In particular, we get an upper exponential bound and construct an analogue to the De Vylder approximation for the ruin probability. We compare results of these approaches with statistical estimates obtained by the Monte Carlo method for selected distributions of claim sizes and additional funds.