Abstract
In this paper, we consider a class of bidimensional discrete-time risk models, which are based on the assumptions that the claim counts obey some specific bivariate integer-valued time series such as bivariate Poisson MA (BPMA) and the bivariate Poisson AR (BPAR) processes. We derive the moment generating functions (m.g.f.’s) for these processes, and we present their explicit expressions for the adjustment coefficient functions. The asymptotic approximations (upper bounds) to three different types of ruin probabilities are discussed, and the marginal value-at-risk (VaR) for each model is obtained. Numerical examples are provided to compute the adjustment coefficients discussed in the paper.
Highlights
Bidimensional risk theory has gained a lot of attention in the last two decades due to its complexity and various uses in different fields
If we regard the claim counts of each one responsibility of the policies as integer-valued time series, and they are correlated, the whole claim counts should be a bivariate integer-valued time series, and the models proposed by Pedeli and Karlis [ ] meet this situation perfectly. We extend their risk models to bidimensional contexts, and we study the bidimensional risk models based on the bivariate claim counts obeying bivariate integer-valued time series
In Section, we present the risk models based on the bivariate claim counts obeying bivariate Poisson MA( ) (BPMA( )) and the bivariate Poisson AR( ) (BPAR( )) process generated by binomial thinning operations
Summary
Bidimensional risk theory has gained a lot of attention in the last two decades due to its complexity and various uses in different fields. Considering the dependent relationship of the claim counts among different periods, the univariate integer-valued time series has been applied to describe it, Cossétte et al [ , ] applied the Poisson MA( ) and Poisson AR( ) processes to discrete-time risk models. We propose a class of general bidimensional risk models based on bivariate time series for the bivariate claim counts r.v.’s. In Section , we present the risk models based on the bivariate claim counts obeying bivariate Poisson MA( ) (BPMA( )) and the bivariate Poisson AR( ) (BPAR( )) process generated by binomial thinning operations. We derive the joint c.g.f., which is denoted by cn(t, s), of the aggregate net losses process based on model BPMA( ).
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