| My thesis focuses on the evaluation of key performance measures for stochastic processes that arise in insurance risk analysis and engineering systems. We are especially interested in scenarios where traditional analytical solutions are unavailable, hence forcing the need to resort to asymptotic approximations and Monte Carlo methods that hang on structural insights on the process dynamics.;Broadly speaking, the first half of my thesis is devoted to processes that build on random walks and their variants, but possess more exotic tail and dependency behaviors that aim to capture real-world features. In particular, we study three related aspects: a central limit refinement for regularly varying random walk beyond the classical so-called Edgeworth expansion, a complete large and moderate deviations result for the maximum of a class of subexponential random walks, and the estimation of the tail probability of perpetuities, or infinite horizon discounted sums that carry random walks in disguise in their exponents. These approximations find applications in insurance ruin analysis, physical modeling and queueing systems.;In the second half, we focus on the tail analysis of a large-scale stochastic system known as many-server queue, arising particularly in communication and call center engineering. In this system, customers arrive, stay, and leave according to presumed general distributions. The system has finite but large number of servers, whose full utilization would lead to the loss of arriving customers. The calculation of the long-run loss probability forms a central question in the performance evaluation of the system. Despite its importance, the gigantic system size and the rare-event nature of loss impose difficulty in obtaining accurate solution. Here we construct provably efficient importance sampling scheme to measure such quantity, employing a novel sample-path approach that requires measure-valued process descriptor, while simultaneously deriving a new large deviations asymptotic result. |
