Title: Exact simulation algorithms with applications in queueing theory and extreme value analysis
Description: This dissertation focuses on the development and analysis of exact simulation algorithms with applications in queueing theory and extreme value analysis. We first introduce the first algorithm that samples max_𝑛≥0 {𝑆_𝑛 − 𝑛^α} where 𝑆_𝑛 is a mean zero random walk, and 𝑛^α with α ∈ (1/2,1) defines a nonlinear boundary. We apply this algorithm to construct the first exact simulation method for the steady-state departure process of a 𝐺𝐼/𝐺𝐼/∞ queue where the service time distribution has infinite mean. Next, we consider the random field 𝑀 (𝑡) = sup_(𝑛≥1) 􏰄{ − log 𝑨_𝑛 + 𝑋_𝑛 (𝑡)􏰅}, 𝑡 ∈ 𝑇 , for a set 𝑇 ⊂ ℝ^𝓂, where (𝑋_𝑛) is an iid sequence of centered Gaussian random fields on 𝑇 and 𝑂 < 𝑨₁ < 𝑨₂ < . . . are the arrivals of a general renewal process on (0, ∞), independent of 𝑋_𝑛. In particular, a large class of max-stable random fields with Gumbel marginals have such a representation. Assume that the number of function evaluations needed to sample 𝑋_𝑛 at 𝑑 locations 𝑡₁, . . . , 𝑡_𝑑 ∈ 𝑇 is 𝑐(𝑑). We provide an algorithm which samples 𝑀(𝑡_{1}), . . . ,𝑀(𝑡_𝑑) with complexity 𝑂 (𝑐(𝑑)^{1+𝘰 (1)) as measured in the 𝐿_𝑝 norm sense for any 𝑝 ≥ 1.

Moreover, if 𝑋_𝑛 has an a.s. converging series representation, then 𝑀 can be a.s. approximated with error δ uniformly over 𝑇 and with complexity 𝑂 (1/(δl og (1/\δ((^{1/α}, where α relates to the Hölder continuity exponent of the process 𝑋_𝑛 (so, if 𝑋_𝑛 is Brownian motion, α =1/2). In the final part, we introduce a class of unbiased Monte Carlo estimators for multivariate densities of max-stable fields generated by Gaussian processes. Our estimators take advantage of recent results on the exact simulation of max-stable fields combined with identities studied in the Malliavin calculus literature and ideas developed in the multilevel Monte Carlo literature. Our approach allows estimating multivariate densities of max-stable fields with precision 𝜀 at a computational cost of order 𝑂 (𝜀 ⁻² log log log 1/𝜀).