Large Deviations Principles on Jackson Network and Importance Sampling for a Jump-Diffusion ProcessSun, WeifengcreatorWang, HuiAdvisorDong, HongjieReaderPirjol, DanReaderBrown University. Department of Applied Mathematicssponsor2019ix, 77 p.born digitalThesis (Ph. D.)--Brown University, 2019thesesLarge deviations are classical topics of study that enjoy great practical application under probability theory. In essence, it studies the properties lie in the tail events of probability measure. In this thesis, we study two problems in both theoretical analyses of large deviation properties and numerical simulations. In the first part of the thesis, we investigate large deviation properties on Jackson networks embedded with preemptive service discipline. This preemptive service discipline introduces the discontinuous dynamics in the system. Thus the classical Cramer’s theorem cannot be utilized. The discontinuous statistics problem generated from preemptive service discipline is not fully studied and enjoys great practical applications. Aim at generating a general large deviation results, we prove the large deviation properties on a very common queuing network -- Jackson Network. The difficulty of this problem lies in showing the large deviation lower bound and possible additional “stability-at-the-interface” condition. In the thesis, we convert the large deviation problem into a stochastic control problem and utilize the weak convergence approach to justify that the “stability-at-the-interface” condition is automatically embedded in the system. The second part of the thesis concentrates on the numerical simulation of tail events. Importance sampling is a classical variance reduction technique and the art of importance sampling lies in the selection of alternative sampling distribution. The tail events are easy to simulate under a “good” alternative distribution. In this study, we select mixtures of exponential tilting distributions as a candidate and establish a cross-entropy iterative algorithm to approximate the optimal one. Then we utilize this technology to pricing some deep out-of-money options and achieve great variance reduction performance. Besides, some large deviation results for the cross-entropy solution are investigated.Monte Carlo methodLarge deviationsProbabilityOptions (Finance)--Prices--Mathematical modelsQueuing theory--Mathematical modelsEnglishRPB2019060310.26300/sn06-te46In CopyrightCollection is open for research.dissertations