| This dissertation focuses on the multivariate compound point processes with drifts and their applications to reliability modeling and financial risk management.; Consider a system that consists of multiple components. The random shocks arrive at the system according to a stochastic point process, and each shock incurs several types of damages, one for each component, that are usually stochastically dependent. As long as the system is working, it generates some rewards, such as income, that can be used in maintenance of the system. The cumulative damages on various components over time can be described by a multivariate compound point process with drifts in which drift rates represent the reward rates. The performance measures of interests are various multivariate ruin/failure probabilities. These ruin/failure probabilities are of fundamental importance to the system operations and management, but there are no closed formulas even in some simplest cases.; We introduce in this dissertation a general multivariate compound point process with drifts, and discuss various ruin probabilities. Utilizing stochastic comparison methods, we analytically compare two such processes with different parameters and obtain some computable bounds for various ruin/failure probabilities that have no closed form expressions. We also apply the results to reliability and risk modeling and show how ignoring dependence among various types of damages would result in over-estimating or underestimating the ruin/failure probabilities. In addition, We utilize multivariate phase-type distributions to model random damages and obtain, via the matrix method, explicit expressions for certain ruin/failure probabilities. The results we obtained are illustrated by numerous examples and extensive stochastic simulations. |
