Appendix to "Approximating perpetuities"

An algorithm for perfect simulation from the unique solution of the distributional fixed point equation $Y=_d UY + U(1-U)$ is constructed, where $Y$ and $U$ are independent and $U$ is uniformly distributed on $[0,1]$. This distribution comes up as a limit distribution in the probabilistic analysis of the Quickselect algorithm. Our simulation algorithm is based on coupling from the past with a multigamma coupler. It has four lines of code

Representation and Simulation of Multivariate Dickman Distributions and Vervaat Perpetuities

A multivariate extension of the Dickman distribution was recently introduced, but very few properties have been studied. We discuss several properties with an emphasis on simulation. Further, we introduce and study a multivariate extension of the more general class of Vervaat perpetuities and derive a number of properties and representations. Most of our results are presented in the even more general context of so-called $\alpha$-times self-decomposable distributions

Return to the Poissonian City

Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial / final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination / source. Suppose further that connections are established using "near-geodesics", constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting near-geodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In earlier work ("Geodesics and flows in a Poissonian city", Annals of Applied Probability, 21(3), 801--842, 2011) it was shown that a scaled version of this flow had asymptotic distribution given by the 4-volume of a region in 4-space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two "seminal curves" defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to an L1 error which tends to zero with increasing computational effort.Comment: 11 pages, 2 figures Various minor edits, corrections to multiplicative constants in Theorem 5.1. Version 2: minor stylistic corrections, added acknowledgement of grant support. Version 3: three further minor corrections. This paper is due to appear in Journal of Applied Probability, Volume 51

Convergence to type I distribution of the extremes of sequences defined by random difference equation

We study the extremes of a sequence of random variables $(R_n)$ defined by the recurrence $R_n=M_nR_{n-1}+q$, $n\ge1$, where $R_0$ is arbitrary, $(M_n)$ are iid copies of a non--degenerate random variable $M$, $0\le M\le1$, and $q>0$ is a constant. We show that under mild and natural conditions on $M$ the suitably normalized extremes of $(R_n)$ converge in distribution to a double exponential random variable. This partially complements a result of de Haan, Resnick, Rootz\'en, and de Vries who considered extremes of the sequence $(R_n)$ under the assumption that $\P(M>1)>0$.Comment: to appear in Stochastic Processes and their Application

Exact simulation of generalised Vervaat perpetuities

We consider a generalised Vervaat perpetuity of the form X = Y 1 W 1 +Y 2 W 1 W 2 + · · ·, where and (Y i) i≥0 is an independent and identically distributed sequence of random variables independent from (W i) i≥0. Based on a distributional decomposition technique, we propose a novel method for exactly simulating the generalised Vervaat perpetuity. The general framework relies on the exact simulation of the truncated gamma process, which we develop using a marked renewal representation for its paths. Furthermore, a special case arises when Y i = 1, and X has the generalised Dickman distribution, for which we present an exact simulation algorithm using the marked renewal approach. In particular, this new algorithm is much faster than existing algorithms illustrated in Chi (2012), Cloud and Huber (2017), Devroye and Fawzi (2010), and Fill and Huber (2010), as well as being applicable to the general payments case. Examples and numerical analysis are provided to demonstrate the accuracy and effectiveness of our method

Exact Simulation of the Extrema of Stable Processes

We exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of L\'evy processes) and apply it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse numerically the performance of the algorithm (the code, written in Julia 1.0, is available on GitHub).Comment: 26 pages, 3 figures, Julia implementation of the exact simulation algorithm is in the GitHub repository: https://github.com/jorgeignaciogc/StableSupremum.j

Approximation algorithms for the normalizing constant of Gibbs distributions

Consider a family of distributions $\{\pi_{\beta}\}$ where $X\sim\pi_{\beta}$ means that $\mathbb{P}(X=x)=\exp(-\beta H(x))/Z(\beta)$. Here $Z(\beta)$ is the proper normalizing constant, equal to $\sum_x\exp(-\beta H(x))$. Then $\{\pi_{\beta}\}$ is known as a Gibbs distribution, and $Z(\beta)$ is the partition function. This work presents a new method for approximating the partition function to a specified level of relative accuracy using only a number of samples, that is, $O(\ln(Z(\beta))\ln(\ln(Z(\beta))))$ when $Z(0)\geq1$. This is a sharp improvement over previous, similar approaches that used a much more complicated algorithm, requiring $O(\ln(Z(\beta))\ln(\ln(Z(\beta)))^5)$ samples.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1015 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Simulations on Lévy subordinators and Lévy driven contagion models

Lévy subordinators have become a fundamental component to be used to construct many useful stochastic processes, which have numerous applications in finance, insurance and many other fields. However, as many applications of Lévy based stochastic models use fairly complicated analytical and probabilistic tools, it has been challenging to implement in practice. Hence, simulation-based study becomes more desirable. In this thesis, we deal with exact simulation on Lévy subordinators and Lévy driven stochastic models. In the first part, we focus on developing more efficient exact simulation schemes for Lévy subordinators with existing simulation algorithms in the literature. Besides, we also introduce a new type of Lévy subordinators, i.e. truncated Lévy subordinators. We study the path properties, develop exact simulation algorithms based on marked renewal representations, and provide relevant applications in finance and insurance. The associated results in this part are later used in the sequel. The second part of this thesis proposes a new type of point processes by generalising the classical self-exciting Hawkes processes and doubly stochastic Poisson processes with Lévy driven Ornstein-Uhlenbeck type intensities. These resulting models are analytically tractable, and intrinsically inherit the great flexibility as well as desirable features from the two original processes, including skewness, leptokurtosis, mean-reverting dynamics, and more importantly, the contagion or feedback effects. These newly constructed processes would then substantially enrich continuous-time models tailored for quantifying the contagion of event arrivals in finance, economics, insurance, queueing and many other fields. In turn, we characterise the distributional properties of this new class of point processes and design an exact simulation algorithm to generate sample paths. This is done by applying the exact distributional decomposition technique. We carry out extensive numerical implementations and tests to demonstrate the accuracy and effectiveness of our scheme and give examples of some financial applications to credit portfolio risk to show the applicability and flexibility of our new model