Smoothed Monte Carlo estimators for the time-in-the-red in risk processes

We consider a modified version of the de Finetti model in insurance risk theory in which, when surpluses become negative the company has the possibility of borrowing, and thus continue its operation. For this model we examine the problem of estimating the time-in-the red over a finite horizon via simulation. We propose a smoothed estimator based on a conditioning argument which is very simple to implement as well as particularly efficient, especially when the claim distribution is heavy tailed. We establish unbiasedness for this estimator and show that its variance is lower than the naïve estimator based on counts. Finally we present a number of simulation results showing that the smoothed estimator has variance which is often significantly lower than that of the naïve Monte-Carlo estimator.     

Range-based Multi-Actor Multi-Criteria Analysis: A combined method of Multi-Actor Multi-Criteria Analysis and Monte Carlo simulation to support participatory decision making under uncertainty

Concerns about environmental and social effects have made Multi-Criteria Decision Making (MCDM) increasingly popular. Decision making in complex contexts often – possibly always – requires addressing an aggregation of multiple issues to meet social, economic, legal, technical, and environmental objectives. These values at stake may affect different stakeholders through distributional effects characterized by a high and heterogeneous uncertainty that no social actors can completely control or understand. On this basis, we present a new process framework that aims to support participatory decision making under uncertainty: the range-based Multi-Actor Multi-Criteria Analysis (range-based MAMCA). On the one hand, the process framework explicitly considers stakeholders’ objectives at an output level of aggregation. On the other hand, by means of a Monte Carlo analysis, the method also provides an exploratory scenario approach that enables the capture of the uncertainty, which stems from the complex context evolution. Range-based MAMCA offers a unique participatory process framework that enables us (1) to identify the alternatives pros and cons for each stakeholder group; (2) to provide probabilities about the risk of supporting mistaken, or at least ill-suited, decisions because of the uncertainty regarding to the decision-making context; (3) to take the decision-makers’ limited control of the actual policy effects over the implementation of one or several options into account. The range-based MAMCA framework is illustrated by means of our first case study that aimed to assess French stakeholders’ support for different biofuel options by 2030.     

蒙特卡罗模拟法在边坡可靠性分析中的运用

江永红．蒙特卡罗模拟法在边坡可靠性分析中的运用．数理统计与管理，１９９８，１７（１），１３～１６．可靠性分析是边坡工程及滑坡治理中的重要研究课题。鉴于决定边坡可靠性的诸变量多为随机变量，本文论述了用蒙特卡罗模拟法计算边坡可靠度的基本原理，对模拟次数确定、计算误差估计等问题提出了解决办法，并结合具体运用说明该方法的实施步骤     

Stochastic Simulation and Monte Carlo methods applied to the assessment of hydro-thermal generating system operation

Summary Simulation can be defined as a numerical technique for conducting experiments on a digital computer, which involves certain types of mathematical and logical models that describe the behaviour of a system over extended periods of real time. Simulation is, in a wide sense, a technique for performing sampling experiments on a model of the system. Stochastic simulation implies experimenting with the model over time including sampling stochastic variates from probability distributions. This paper describes the main concepts of the application of Stochastic Simulation and Monte Carlo methods to the analysis of the operation of electric energy systems, in particular to hydro-thermal generating systems. These techniques can take into account virtually all contingencies inherent in the operation of the system. Also, the operating policies that have an important effect on the performance of these systems can be realistically represented.     

Weighted Monte Carlo method for an approximate solution of the nonlinear coagulation equation

New weighted modifications of direct statistical simulation methods designed for the approximate solution of the nonlinear Smoluchowski equation are developed on the basis of stratification of the interaction distribution in a multiparticle system according to the index of a pair of interacting particles. The weighted algorithms are validated for a model problem with a known solution. It is shown that they effectively estimate variations in the functionals with varying parameters, in particular, with the initial number N ₀ of particles in the simulating ensemble. The computations performed for the problem with a known solution confirm the semiheuristic hypothesis that the model error is O(N ₀ ^?1 ). Estimates are derived for the derivatives of the approximate solution with respect to the coagulation coefficient.     

Discrete stochastic consistent estimators of the Monte Carlo method

The efficiency of discrete stochastic consistent estimators (the weighted uniform sampling and estimator with a correcting multiplier) of the Monte Carlo method is investigated. Confidence intervals and upper bounds on the variances are obtained, and the computational cost of the corresponding discrete stochastic numerical scheme is estimated.     

Dominant eigenvalue problem for positive integral operators and its solution by Monte Carlo method

In this paper, a method of numerical solution to the dominant eigenvalue problem for positive integral operators is presented. This method is based on results of the theory of positive operators developed by Krein and Rutman. The problem is solved by Monte Carlo method constructing random variables in such a way that differences between results obtained and the exact ones would be arbitrarily small. Some numerical results are shown.     

Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation

We construct weighted modifications of statistical modeling of an ensemble of interacting particles which is connected with approximate solution of a nonlinear Boltzmann equation.     

Conditional Monte Carlo method for dynamic systems with random properties

A method is developed for calculating moments and other properties of states X(t) of dynamic systems with random coefficients depending on semi-Markov processes ξ(t) and subjected to Gaussian white noise. Random vibration theory is used to find probability laws of conditional processes X(t)∣ξ(·). Unconditional properties of X(t) are estimated by averaging conditional statistics of this process corresponding to samples of ξ(t). The method is particularly efficient for linear systems since X(t)∣ξ(·) is Gaussian during periods of constant values of ξ(t), so that and its probability law is completely defined by the process mean and covariance functions that can be obtained simply from equations of linear random vibration. The method is applied to find statistics of an Ornstein-Uhlenbeck process X(t) whose decay parameter is a semi-Markov process ξ(t). Numerical results show that X(t) is not Gaussian and that the law of this process depends essentially on features of ξ(t). A version of the method is used to calculate the failure probability for an oscillator with degrading stiffness subjected to Gaussian white noise.     

On the adoption of the Monte Carlo method to solve one-dimensional steady state thermal diffusion problems for non-uniform solids

The present paper is focussed on the investigation of the potential adoption of the Monte Carlo method to solve one-dimensional, steady state, thermal diffusion problems for continuous solids characterised by an isotropic, space-dependent conductivity tensor and subjected to non-uniform heat power deposition.     

The Monte Carlo EM method for estimating multinomial probit latent variable models

We propose a multinomial probit (MNP) model that is defined by a factor analysis model with covariates for analyzing unordered categorical data, and discuss its identification. Some useful MNP models are special cases of the proposed model. To obtain maximum likelihood estimates, we use the EM algorithm with its M-step greatly simplified under Conditional Maximization and its E-step made feasible by Monte Carlo simulation. Standard errors are calculated by inverting a Monte Carlo approximation of the information matrix using Louis’s method. The methodology is illustrated with a simulated data.     

Ant colony method to control variance reduction techniques in the Monte Carlo simulation of clinical electron linear accelerators of use in cancer therapy

The Monte Carlo simulation of clinical electron linear accelerators requires large computation times to achieve the level of uncertainty required for radiotherapy. In this context, variance reduction techniques play a fundamental role in the reduction of this computational time. Here we describe the use of the ant colony method to control the application of two variance reduction techniques: Splitting and Russian roulette. The approach can be applied to any accelerator in a straightforward way and permits the increasing of the efficiency of the simulation by a factor larger than 50.     

Asymptotics of an Efficient Monte Carlo Estimation for the Transition Density of Diffusion Processes

Discretized simulation is widely used to approximate the transition density of discretely observed diffusions. A recently proposed importance sampler, namely modified Brownian bridge, has gained much attention for its high efficiency relative to other samplers. It is unclear for this sampler, however, how to balance the trade-off between the number of imputed values and the number of Monte Carlo simulations under a given computing resource. This paper provides an asymptotically efficient allocation of computing resource to the importance sampling approach with a modified Brownian bridge as importance sampler. The optimal trade-off is established by investigating two types of errors: Euler discretization error and Monte Carlo error. The main results are illustrated with two simulated examples.      

Modifications of weighted Monte Carlo algorithms for nonlinear kinetic equations

Test problems for the nonlinear Boltzmann and Smoluchowski kinetic equations are used to analyze the efficiency of various versions of weighted importance modeling as applied to the evolution of multiparticle ensembles. For coagulation problems, a considerable gain in computational costs is achieved via the approximate importance modeling of the “free path” of the ensemble combined with the importance modeling of the index of a pair of interacting particles. A weighted modification of the modeling of the initial velocity distribution was found to be the most efficient for model solutions to the Boltzmann equation. The technique developed can be useful as applied to real-life coagulation and relaxation problems for which the model problems considered give approximate solutions.     

Corrections to the Article “Weighted Monte Carlo Methods for Approximate Solution of a Nonlinear Boltzmann Equation”

Corrections are given to the above-mentioned article.     

Monte Carlo construction of hedging strategies against multi-asset European claims

For evaluating a hedging strategy we have to know at every moment the solution of the Cauchy problem for a corresponding parabolic equation (the value of the hedging portfolio) and its derivatives (the deltas). We suggest to find these quantities by Monte Carlo simulation of the corresponding system of stochastic differential equations using weak solution schemes. It turns out that with one and the same control function a variance reduction can be achieved simultaneously for the claim value as well as for the deltas. As illustrations we consider a Markovian multi-asset model with an instantaneously riskless saving bond and also some applications to the LIBOR rate model of Brace, Gatarck, Musiela and Jamshidian.     

Efficient implementation of the Barnes-Hut octree algorithm for Monte Carlo simulations of charged systems

Computer simulation with Monte Carlo is an important tool to investigate the function and equilibrium properties of many biological and soft matter materials solvable in solvents.The appropriate treatment of long-range electrostatic interaction is essential for these charged systems,but remains a challenging problem for large-scale simulations.We develop an efficient Barnes-Hut treecode algorithm for electrostatic evaluation in Monte Carlo simulations of Coulomb many-body systems.The algorithm is based on a divide-and-conquer strategy and fast update of the octree data structure in each trial move through a local adjustment procedure.We test the accuracy of the tree algorithm,and use it to perform computer simulations of electric double layer near a spherical interface.It is shown that the computational cost of the Monte Carlo method with treecode acceleration scales as log N in each move.For a typical system with ten thousand particles,by using the new algorithm,the speed has been improved by two orders of magnitude from the direct summation.     

Robustness and applicability of Markov chain Monte Carlo algorithms for eigenvalue problems

In this paper we analyse applicability and robustness of Markov chain Monte Carlo algorithms for eigenvalue problems. We restrict our consideration to real symmetric matrices.

Almost Optimal Monte Carlo (MAO) algorithms for solving eigenvalue problems are formulated. Results for the structure of both – systematic and probability error are presented. It is shown that the values of both errors can be controlled independently by different algorithmic parameters. The results present how the systematic error depends on the matrix spectrum. The analysis of the probability error is presented. It shows that the close (in some sense) the matrix under consideration is to the stochastic matrix the smaller is this error. Sufficient conditions for constructing robust and interpolation Monte Carlo algorithms are obtained. For stochastic matrices an interpolation Monte Carlo algorithm is constructed.

A number of numerical tests for large symmetric dense matrices are performed in order to study experimentally the dependence of the systematic error from the structure of matrix spectrum. We also study how the probability error depends on the balancing of the matrix.     

Efficient Monte Carlo simulation for integral functionals of Brownian motion

In the paper, we develop a variance reduction technique for Monte Carlo simulations of integral functionals of a Brownian motion. The procedure is based on a new method of sampling the process, which combines the Brownian bridge construction with conditioning on integrals along paths of the process. The key element in our method is the identification of a low-dimensional vector of variables that reduces the dimension of the integration problem more effectively than the Brownian bridge. We illustrate the method by applying it in conjunction with low-discrepancy sequences to the problem of pricing Asian options.     

Comparing simulation models for market risk stress testing

The subprime crisis has reminded us that effective stress tests should not only combine subjective scenarios with historical data, but also be probabilistic. In this paper, we combine three hypothetical shocks, of varying degrees, with more than six years of daily data on USD-INR and Euro-INR. Our objective is to compare six simulation-based stress models for foreign exchange positions. We find that while volatility-weighted historical simulation is the best model for volatility persistence, jump diffusion based Monte Carlo simulation is better at capturing correlation breakdown. Loss estimates from very fat-tailed distributions are not sensitive to the severity of stress scenarios.