Motivated by recent developments of quasi-stationary Monte Carlo methods, we investigate the stability of quasi-stationary distributions of killed Markov processes under perturbations of the generator. We first consider a general bounded self-adjoint perturbation operator, and then study a particular unbounded perturbation corresponding to truncation of the killing rate. In both scenarios, we quantify the difference between eigenfunctions of the smallest eigenvalue of the perturbed and unperturbed generators in a Hilbert space norm. As a consequence, \mathcal{L}^1$\mathcal{L}^1$-norm estimates of the difference of the resulting quasi-stationary distributions in terms of the perturbation are provided.

An algorithm for generating artificial data sets which contain distinct nonoverlapping clusters is presented. The algorithm is useful for generating test data sets for Monte Carlo validation research conducted on clustering methods or statistics. The algorithm generates data sets which contain either 1, 2, 3, 4, or 5 clusters. By default, the data are embedded in either a 4, 6, or 8 dimensional space. Three different patterns for assigning the points to the clusters are provided. One pattern assigns the points equally to the clusters while the remaining two schemes produce clusters of unequal sizes. Finally, a number of methods for introducing error in the data have been incorporated in the algorithm.

Rasch proposed an exact conditional inference approach to testing his model but never implemented it because it involves the calculation of a complicated probability. This paper furthers Rasch’s approach by (1) providing an efficient Monte Carlo methodology for accurately approximating the required probability and (2) illustrating the usefulness of Rasch’s approach for several important testing problems through simulation studies. Our Monte Carlo methodology is shown to compare favorably to other Monte Carlo methods proposed for this problem in two respects: it is considerably faster and it provides more reliable estimates of the Monte Carlo standard error.

Data in the form of zero-one matrices where conditioning on the marginals is relevant arise in diverse fields such as social networks and ecology; directed graphs constitute an important special case. An algorithm is given for the complete enumeration of the family of all zero-one matrices with given marginals and with a prespecified set of cells with structural zero entries. Complete enumeration is computationally feasible only for relatively small matrices. Therefore, a more useable Monte Carlo simulation method for the uniform distribution over this family is given, based on unequal probability sampling and ratio estimation. This method is applied to testing reciprocity of choices in social networks.

The basic theme of the EM algorithm, to repeatedly use complete-data methods to solve incomplete data problems, is also a theme of several more recent statistical techniques. These techniques—multiple imputation, data augmentation, stochastic relaxation, and sampling importance resampling—combine simulation techniques with complete-data methods to attack problems that are difficult or impossible for EM.

We establish a pathwise large deviation principle for affine stochastic volatility models introduced by Keller-Ressel (2011), and present an application to variance reduction for Monte Carlo computation of prices of path-dependent options in these models, extending the method developed by Genin and Tankov (2020) for exponential Lévy models. To this end, we apply an exponentially affine change of measure and use Varadhan’s lemma, in the fashion of Guasoni and Robertson (2008) and Robertson (2010), to approximate the problem of finding the measure that minimizes the variance of the Monte Carlo estimator. We test the method on the Heston model with and without jumps to demonstrate its numerical efficiency.

A new theoretical method for examining gerrymandering is presented based on lattice models of voters, in which districts are constructed by partitioning the lattice. We propose three novel algorithms for constructing equal-population, connected districts which favor the gerrymanderer and incorporate the spatial distribution of voters. Due to the probabilistic population fluctuations inherent to our voter models, Monte Carlo techniques can be applied to study the impact of gerrymandering. We use the method developed here to compare our different gerrymandering algorithms, show approaches which ignore spatial data lead to (legally prohibited) disconnected districts, and examine the effectiveness of isoperimetric quotient tests.

Political scientists are increasingly interested in the geographic distribution of political and economic phenomena. Unlike distribution measures at the individual level, geographic distributions depend on the “unit question” in which researchers choose the appropriate political subdivision to analyze, such as nations, subnational regions, urban and rural areas, or electoral districts. We identify concerns with measuring geographic distribution and comparing distributions within and across political units. In particular, we highlight the potential for threats to inference based on the modifiable areal unit problem (MAUP), whereby measuring concepts at different unit aggregations alters the observed value. We offer tangible options for researchers to improve their research design and data analysis to limit the MAUP. To help manage the measurement error when the unit of observation is unclear or appropriate data are not available, we introduce a new measure of geographic distribution that accounts for fluctuations in the scale and number of political units considered. We demonstrate using Monte Carlo simulations that our measure is more reliable and stable across political units than commonly used indicators because it reduces measurement fluctuations associated with the MAUP.

This paper presents comprehensive studies on two closely related problems of high speed collisionless gaseous jet from a circular exit and impinging on an inclined rectangular flat plate, where the plate surface can be diffuse or specular reflective. Gaskinetic theories are adopted to study the problems, and several crucial geometry-location and velocity-direction relations are used. The final complete results include flowfield properties such as density, velocity components, temperature and pressure, and impingement surface properties such as coefficients of pressure, shear stress and heat flux. Also included are the averaged coefficients for pressure, friction, heat flux, moment over the whole plate, and the averaged distance from the moment center to the plate center. The final results include complex but accurate integrations involving the geometry and specific speed ratios, inclination angle, and the temperature ratio. Several numerical simulations with the direct simulation Monte Carlo method validate these analytical results, and the results are essentially identical. Exponential, trigonometric, and error functions are embedded in the solutions. The results illustrate that the past simple cosine function approach is rather crude, and should be used cautiously. The gaskinetic method and processes are heuristic and can be used to investigate other external high Knudsen number impingement flow problems, including the flowfield and surface properties for high Knudsen number jet from an exit and flat plate of arbitrary shapes. The results are expected to find many engineering applications.

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for non-linear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully non-linear case and open research questions.

In their tubelike phase, nanowire-adsorbed polymers exhibit strong structural similarities to morphologies known from single-walled carbon (hexagonal) and boron (triangular) nanotubes. Since boron/boron nitride tubes require some disorder for stability the triangular polymer tubes provide a closer analog to the carbon tubes. By means of computer simulations of both two and three dimensional versions of a coarse-grained bead-spring model for the polymers, we investigate their structural properties and make a detailed comparison with structures of carbon nanotubes.

A simulation investigation of the effect of default insurance on the optimal equity allocation and deficit spread period of a model defined benefit pension scheme is performed, using the old and new frameworks of the Pension Protection Fund in the U.K. as a starting point. The old default insurance levy framework encourages an increase in the allocation to equities, creating an indirect effect of increased deficits. The new framework reverses the effect to a reduction in the allocation to equities, thus reducing deficits. In addition the gaming element of default insurance is investigated and found to significantly increase optimal equity allocation and deficit spread period, leading to a significant increase in deficits.

Simulations of a model pension scheme are run with stochastic economic and demographic factors, with an aim to investigate the impact of these factors on movements in funding ratio and average contribution rates. These impacts are analysed by running regressions of movements in funding ratio and average contribution rates against the economic and demographic factors. It is found that, for a typical scheme closed to new entrants and a balanced asset allocation including equity investment, the mismatch between discount rate movements and investment returns is by far the biggest predictor of funding ratio movements, with average contribution rates affected more by events in a few individual years rather than averaged over an entire simulation. Where the scheme invests to cash-flow match liabilities, mortality improvement becomes the most significant predictor of funding ratio movements, although mortality improvement still has little impact on average contribution rates.

A new criterion is presented to detect global convergence to steady state, and to identify local transient characteristics, during rarefied gas flow simulations performed using the direct simulation Monte Carlo (DSMC) method. Unlike deterministic computational fluid dynamics (CFD) schemes, DSMC is generally subject to large statistical scatter in instantaneous flow property evaluations, which prevents the use of residual tracking procedures as are often employed in CFD simulations. However, reliable prediction of the time to reach steady state is necessary for initialization of DSMC sampling operations. Techniques currently used in DSMC to identify steady state convergence are usually insensitive to weak transient behavior in small regions of relatively low density or recirculating flow. The proposed convergence criterion is developed with the goal of properly identifying such weak transient behavior, while adding negligible computational expense and allowing simple implementation in any existing DSMC code. Benefits of the proposed technique over existing convergence detection methods are demonstrated for representative nozzle/plume expansion flow, hypersonic blunt body flow and driven cavity flow problems.

When analyzing catastrophic risk, traditional measures for evaluating risk, such as the probable maximum loss (PML), value at risk (VaR), tail-VaR, and others, can become practically impossible to obtain analytically in certain types of insurance, such as earthquake, and certain types of reinsurance arrangements, specially non-proportional with reinstatements. Given the available information, it can be very difficult for an insurer to measure its risk exposure. The transfer of risk in this type of insurance is usually done through reinsurance schemes combining diverse types of contracts that can greatly reduce the extreme tail of the cedant's loss distribution. This effect can be assessed mathematically. The PML is dei ned in terms of a very extreme quantile. Also, under standard operating conditions, insurers use several “layers” of non proportional reinsurance that may or may not be combined with some type of proportional reinsurance. The resulting reinsurance structures will then be very complicated to analyze and to evaluate their mitigation or transfer effects analytically, so it may be necessary to use alternative approaches, such as Monte Carlo simulation methods. This is what we do in this paper in order to measure the effect of a complex reinsurance treaty on the risk profile of an insurance company. We compute the pure risk premium, PML as well as a host of results: impact on the insured portfolio, risk transfer effect of reinsurance programs, proportion of times reinsurance is exhausted, percentage of years it was necessary to use the contractual reinstatements, etc. Since the estimators of quantiles are known to be biased, we explore the alternative of using an Extreme Value approach to complement the analysis.

With the pioneering work of [Pardoux and Peng, Syst. Contr. Lett.14 (1990) 55–61; Pardoux and Peng,Lecture Notes in Control and Information Sciences176 (1992) 200–217]. We have at our disposalstochastic processes which solve the so-called backward stochasticdifferential equations. These processes provide us with a Feynman-Kacrepresentation for the solutions of a class of nonlinear partial differential equations (PDEs) which appearin many applications in the field of Mathematical Finance. Therefore thereis a great interest among both practitioners and theoreticians to developreliable numerical methods for their numerical resolution. In this survey, we present a number of probabilistic methods forapproximating solutions of semilinear PDEs all based on the correspondingFeynman-Kac representation. We also include a general introduction tobackward stochastic differential equations and their connection with PDEsand provide a generic framework that accommodates existing probabilisticalgorithms and facilitates the construction of new ones.

We consider estimating tail events using exponential families of importance sampling distributions. When the cannonical sufficient statistic for the exponential family mimics the tail behaviour of the underlying cumulative distribution function, we can achieve bounded relative error for estimating tail probabilities. Examples of rare event simulation from various distributions including Tukey's g&h distribution are provided.

This paper presents a performance comparison of different estimation and prediction techniques applied to the problem of tracking multiple robots. The main performance criteria are the magnitude of the estimation or prediction error, the computational effort and the robustness of each method to non-Gaussian noise. Among the different techniques compared are the well-known Kalman filters and their different variants (e.g. extended and unscented), and the more recent techniques relying on Sequential Monte Carlo Sampling methods, such as particle filters and Gaussian Mixture Sigma Point Particle Filter.