Analysis Of Some Risk Problems

Risk exists in all our life. Statistics and probability is one of the best tools to study the risk. This paper gives applications of statistics in the analysis of fire risks and study the problem of optimal portfolio.With the developing of the economy, fire risk is more and more important. In this paper, cluster and distinguish analyses are used in fire detection. The productions of different materials being combusted is clustered. Because we didn't have the actual fire detection data, distinguish analysis and logistic regression as a likely method to calculate the fire probability are used. For fire forecasting, the suitable criteria for the GM(1,1) is given. On the other hand, we investigate the heavy-tailed property and self-similarity of fire loss data, and use the truncated power-tail (TPT) distribution to model fire loss data.In the theory of portfolio, we first consider the dependent default risk model of factor type. The dependence between the returns of assets is driven by default indicators. Sufficient conditions on the dependence structure of default indicators and on the utility function are investigated which enable one to order the optimal amount invested in each asset. For the optimal allocation of policy limits and deductibles, by applying the bivariate characterizations of stochastic ordering relations, we reconsider the same model and derive some new refined results on orderings of optimal allocations of policy limits and deductibles with respect to the family of distortion risk measures from the viewpoint of the policyholder. Both loss severities and loss frequencies are considered. Special attention is given to the optimization criteria of the family of distortion risk measures with concave distortions and with only increasing distortions. Most of results presented in this paper can be applied to some particular distortion risk measures. The main results extend and strengthen the known results in the literature.