Techniques and Applications for the Efficient Simulation of Gaussian Rare Events
Birge, Richard Gabriel
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This dissertation focuses on rare-event simulation in the context of Gaussian random processes, which are in widespread use amongst industry practitioners. In particular, Gaussian processes have been used in financial disciplines such as risk management and portfolio valuation, because these processes are computationally tractable and come with an abundance of vendor tools. The present work provides both a theoretical and practical treatment of simulating Gaussian rare events, from algorithm construction and asymptotics, to empirical analysis via computational studies across multiple problem instances. The preliminary chapters of this dissertation provide the mathematical framework and theoretical justification for our methodology, including a case study in a simplified problem setting; this treatment is then extended to the later chapters, wherein we generalize previous theory to broader classes of functionals, and also consider specific applications of our method. First, in Chapters 1 and 2, we introduce the research topic area, mathematical context and goals. We give further detail in Chapter 1 on our problem motivation, which is the design of an efficient, generalizable Monte Carlo algorithm for simulating exceedance probabilities of rare-event-type functionals, such as max_i X_i, for discrete Gaussian random vectors X. We follow this introductory chapter with a discussion in Chapter 2 of the mathematical foundations for our proposed simulation method, including a background to importance sampling theory and mathematical notions of algorithmic complexity and efficiency, particularly in asymptotic settings. We analyze two small problem instances where standard variance-reduction tools have computational setbacks; we outline the subtleties that make these problems non-trivial, and show their relevance for the alternative methodology explored in this dissertation. To provide additional understanding for how our method operates and the underlying mathematical intuition, in Chapter 3 we give an expository study focusing on a special problem instance of the constrained maximum functional. In this chapter, we detail the construction of an efficient Monte Carlo estimator wherein the underlying Gaussian vector has standard marginals, for which the minimum is also restricted to be non-negative. Through the estimation problem we study in Chapter 3, we provide the key principles and performance insights for our simulation algorithm, which we highlight through several computational studies. The latter parts of this dissertation, Chapters 4 and 5, extend the earlier theoretical framework to cover a wide class of functions, taken with respect to Gaussian marginals having general variance structures, and also including conditional functionals relevant to financial risk management. In Chapter 4, we give the construction of a general, efficient Monte Carlo estimator to compute exceedance probabilities for rare-event-type functionals of discrete Gaussian vectors, taken with nonstandard marginals, for what we show to be a broad class of functions. The mathematical treatment in Chapter 4 serves to bridge the theory from the special case of the constrained maximum in Chapter 3 to a general abstract problem setting. We provide proofs in Chapter 4 that our general estimator is unbiased and also has desirable asymptotic properties. We substantiate these claims with detailed computational studies. Lastly, in Chapter 5, we present an additional extension to types of functionals relevant to applied settings, for example to assess financial risk. Focusing on topics important to simulation practitioners, particularly the computation of certain risk measures such as expected shortfall or conditional value-at-risk, we describe the usefulness of our method for wide problem applications. We close Chapter 5 with numerical examples that highlight the potential practical uses and flexibility of our simulation method.