In this paper an exact rejection algorithm for simulating paths of the coupled Wright–Fisher diffusion is introduced. The coupled Wright–Fisher diffusion is a family of multivariate Wright–Fisher diffusions that have drifts depending on each other through a coupling term and that find applications in the study of networks of interacting genes. The proposed rejection algorithm uses independent neutral Wright–Fisher diffusions as candidate proposals, which are only needed at a finite number of points. Once a candidate is accepted, the remainder of the path can be recovered by sampling from neutral multivariate Wright–Fisher bridges, for which an exact sampling strategy is also provided. Finally, the algorithm’s complexity is derived and its performance demonstrated in a simulation study.

Coupling-from-the-past (CFTP) methods have been used to generate perfect samples from finite Gibbs hard-sphere models, an important class of spatial point processes consisting of a set of spheres with the centers on a bounded region that are distributed as a homogeneous Poisson point process (PPP) conditioned so that spheres do not overlap with each other. We propose an alternative importance-sampling-based rejection methodology for the perfect sampling of these models. We analyze the asymptotic expected running time complexity of the proposed method when the intensity of the reference PPP increases to infinity while the (expected) sphere radius decreases to zero at varying rates. We further compare the performance of the proposed method analytically and numerically with that of a naive rejection algorithm and of popular dominated CFTP algorithms. Our analysis relies upon identifying large deviations decay rates of the non-overlapping probability of spheres whose centers are distributed as a homogeneous PPP.

There are two types of tempered stable (TS) based Ornstein–Uhlenbeck (OU) processes: (i) the OU-TS process, the OU process driven by a TS subordinator, and (ii) the TS-OU process, the OU process with TS marginal law. They have various applications in financial engineering and econometrics. In the literature, only the second type under the stationary assumption has an exact simulation algorithm. In this paper we develop a unified approach to exactly simulate both types without the stationary assumption. It is mainly based on the distributional decomposition of stochastic processes with the aid of an acceptance–rejection scheme. As the inverse Gaussian distribution is an important special case of TS distribution, we also provide tailored algorithms for the corresponding OU processes. Numerical experiments and tests are reported to demonstrate the accuracy and effectiveness of our algorithms, and some further extensions are also discussed.

We provide the first generic exact simulation algorithm for multivariate diffusions. Current exact sampling algorithms for diffusions require the existence of a transformation which can be used to reduce the sampling problem to the case of a constant diffusion matrix and a drift which is the gradient of some function. Such a transformation, called the Lamperti transformation, can be applied in general only in one dimension. So, completely different ideas are required for the exact sampling of generic multivariate diffusions. The development of these ideas is the main contribution of this paper. Our strategy combines techniques borrowed from the theory of rough paths, on the one hand, and multilevel Monte Carlo on the other.

In this paper, we introduce a new large family of Lévy-driven point processes with (and without) contagion, by generalising the classical self-exciting Hawkes process and doubly stochastic Poisson processes with non-Gaussian Lévy-driven Ornstein–Uhlenbeck-type intensities. The resulting framework may possess many desirable features such as skewness, leptokurtosis, mean-reverting dynamics, and more importantly, the ‘contagion’ or feedback effects, which could be very useful for modelling event arrivals in finance, economics, insurance, and many other fields. We characterise the distributional properties of this new class of point processes and develop an efficient sampling method for generating sample paths exactly. Our simulation scheme is mainly based on the distributional decomposition of the point process and its intensity process. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of our scheme. Moreover, we use portfolio risk management as an example to show the applicability and flexibility of our algorithms.

We present the first algorithm that samples maxn≥0{Sn − nα}, where Sn is a mean zero random walk, and nα with $\alpha \in ({1 \over 2},1)$ defines a nonlinear boundary. We show that our algorithm has finite expected running time. We also 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.

We consider a generalised Vervaat perpetuity of the form X = Y1W1 +Y2W1W2 + · · ·, where $W_i \sim {\cal U}^{1/t}$ and (Yi)i≥0 is an independent and identically distributed sequence of random variables independent from (Wi)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 Yi = 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.

We introduce a new class of Monte Carlo methods, which we call exact estimation algorithms. Such algorithms provide unbiased estimators for equilibrium expectations associated with real-valued functionals defined on a Markov chain. We provide easily implemented algorithms for the class of positive Harris recurrent Markov chains, and for chains that are contracting on average. We further argue that exact estimation in the Markov chain setting provides a significant theoretical relaxation relative to exact simulation methods.

We present an exact simulation algorithm for the stationary distribution of the customer delay D for first-in–first-out (FIFO) M/G/c queues in which ρ=λ/μ<1. We assume that the service time distribution G(x)=P(S≤x),x≥0 (with mean 0<E(S)=1/μ<∞), and its corresponding equilibrium distribution Ge(x)=μ∫0x P(S>y)dy are such that samples of them can be simulated. We further assume that G has a finite second moment. Our method involves the general method of dominated coupling from the past (DCFTP) and we use the single-server M/G/1 queue operating under the processor sharing discipline as an upper bound. Our algorithm yields the stationary distribution of the entire Kiefer–Wolfowitz workload process, the first coordinate of which is D. Extensions of the method to handle simulating generalized Jackson networks in stationarity are also remarked upon.

Our objective is to construct a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structures, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. A tail-lightness condition turns out to be of importance for the applicability of the perfect simulation algorithm. Examples of applications and empirical results are presented.

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.