We consider the wave equation in two spatial dimensions driven by space-time Gaussian noise that is white in time but has a nondegenerate spatial covariance. We give a necessary and sufficient integral condition on the covariance function of the noise for the solution to the linear form of the equation to be a real-valued stochastic process, rather than a distributionvalued random variable. When this condition is satisfied, we show that not only the linear form of the equation, but also nonlinear versions, have a real-valued process solution. We give stronger sufficient conditions on the spatial covariance for the solution of the linear equation to be continuous, and we provide an estimate of its modulus of continuity.

The majority of optimal Bonus-Malus Systems (BMS) presented up to now in the actuarial literature assign to each policyholder a premium based on the number of his accidents. In this way a policyholder who had an accident with a small size of loss is penalized unfairly in the same way with a policyholder who had an accident with a big size of loss. Motivated by this, we develop in this paper, the design of optimal BMS with both a frequency and a severity component. The optimal BMS designed are based both on the number of accidents of each policyholder and on the size of loss (severity) for each accident incurred. Optimality is obtained by minimizing the insurer's risk. Furthermore we incorporate in the above design of optimal BMS the important a priori information we have for each policyholder. Thus we propose a generalised BMS that takes into consideration simultaneously the individual's characteristics, the number of his accidents and the exact level of severity for each accident.

This paper presents the design of optimal Bonus-Malus Systems (BMS) using …nite mixture models, extending the work of Lemaire (1995) and Frangos and Vrontos (2001). Speci…cally, for the frequency component we employ a …nite Poisson, Delaporte and Negative Binomial mixture, while for the severity component we employ a …nite Exponential, Gamma, Weibull and Generalized Beta Type II mixture, updating the posterior probability. We also consider the case of a …nite Negative Binomial mixture and a …nite Pareto mixture updating the posterior mean. The generalized BMS we propose, integrate risk classi…cation and experience rating by taking into account both the a priori and a posteriori characteristics of each policyholder.

We construct generalized weighted Wiener chaos solutions for hyperbolic linear SPDEs driven by a cylindrical Brownian motion. Explicit conditions for the existence, uniqueness, and regularity of generalized (Wiener Chaos) solutions are established in Sobolev spaces. An equivalence relation between the Wiener Chaos solution and the traditional one is established. The Heath-Jarrow-Morton (HJM)forward rate model is used as an example to illustrate the general construction.

An exponential-inverse Gaussian distribution is used to model the claim size distribution. The distribution has shorter tails than the Pareto distribution and it is considered as a plausible model for data without large tails. We present the model allowing for covariates. Properties of the model are discussed. An EM algorithm is provided to fit the model. The algorithm is quite simple and programmable without need for any special functions. The model can be seen as a random effect model for exponential survival times regression. A real data application using a car-insurance company portfolio data is provided.