Η συσχέτιση του πλεονάσματος πριν και μετά τη χρεοκοπία

Γαλανάκης, Πέτρος Χ.

Master Thesis

Author

Γαλανάκης, Πέτρος Χ.

Date

2012-05-15

Abstract

For the classical model with Poisson arrivals of Risk Theory, we study the covariance and correlation of the surplus prior to ruin and the deficit at ruin, given that ruin occurs. When the claim sizes are exponentially distributed it is known that these two variables are stochastically independent. This seems to have a reasonable explanation, due to the fact that the failure rate of equilibrium distribution is not only an increasing but also a decreasing function, at the same time. Since that distribution is Exponential too, as a result the classic model of Risk Theory is bound in "memoryless" property of the Exponential distribution. What happens, however, when the severity distribution only belongs to one aging class? An example, which responds to that question, is the case in which severity distribution is the Gamma. Then for shape parameter bellow one (n < 1), then the failure rate of equilibrium is only increasing. So, that distribution belongs to IFR old age class. However, if the shape parameter is above one (n > 1), then it is decreasing. So, both severity and equilibrium distribution belong to the DFR class. A more advanced example, which is considered in the present work, is the case in which the failure rate changes its monotonicity on the positive axis. This occurs when the severity distribution is a mixture of IFR distributions under some instances, such as mixture of Exponential and Gamma distributions with shape parameter longer the two (n > 2). In that example it is observed that the Exponential distribution is not the only distribution in which the two random variables (depict and surplus) are stochastically independent. In order to be more clear, in this master's thesis we study the correlation between the proper random variables of deficit and surplus (SSEP) for a variety of severity distributions which belong to different aging classes class. It is also studied both the prelementary functions of SSEP and the monotonicity of failure rate of severity distributions, since it is proved that the sign of covariance depends on monotonicity of failure rate.