An Importance Sampling Method for Expectation of Portfolio Credit Risk

The paper is dealing with estimation of rare event probabilities in stochastic networks. The well known variance reduction technique, called Importance Sampling (IS) is an effective tool for doing this. The main idea of IS is to simulate the random system under a modified set of parameters, so as to make the occurrence of the rare event more likely. The major problem of the IS technique is that the optimal modified parameters, called reference parameters to be used in IS are usually very difficult to obtain. Rubinstein (Eur J Oper Res 99:89–112, 1997) developed the Cross Entropy (CE) method for the solution of this problem of IS technique and then he and his collaborators applied this for estimation of rare event probabilities in stochastic networks with exponential distribution [see De Boer et al. (Ann Oper Res 134:19–67, 2005)]. In this paper, we test this simulation technique also for medium sized stochastic networks and compare its effectiveness to the simple crude Monte Carlo (CMC) simulation. The effectiveness of a variance reduction simulation algorithm is measured in the following way. We calculate the product of the necessary CPU time and the estimated variance of the estimation. This product is compared to the same for the simple Crude Monte Carlo simulation. This was originally used for comparison of different variance reduction techniques by Hammersley and Handscomb (Monte Carlo Methods. Methuen & Co Ltd, London, 1967). The main result of the paper is the extension of CE method for estimation of rare event probabilities in stochastic networks with beta distributions. In this case the calculation of reference parameters of the importance sampling distribution requires numerical solution of a nonlinear equation system. This is done by applying a Newton–Raphson iteration scheme. In this case the CPU time spent for calculation of the reference parameter values cannot be neglected. Numerical results will also be presented.

Command-based importance sampling for statistical model checking

At the multiplexer of an ATM network, the admissible cell loss probability is less than 10−12 for some service categories. Ordinary Monte Carlo (MC) simulation requires considerable computation time to obtain such a small probability. The importance sampling (IS) technique is effective in accclerating the MC simulation. In an IS simulation, the underlying distribution is biased so as to generate more samples in the target event. In order to achieve a fast simulation, it is important to have a simulation distribution that yields an estimate with minimum variance. This is called the optimal simulation distribution. In this paper, we investigate the survivor function P(Q > q) of a stationary queue length Q in ATM queuing and obtain an IS estimate of P(Q > q) at about 10−2 by means of the optimal simulation distribution. Conventional methods [1, 2] for determining the optimal distribution require a lot of computation time. We apply the large deviation theory to this problem to obtain the optimal simulation distribution analytically and carry out an IS simulation. We present some numerical results for M/D/1 and 2-state MMPP/D/1 which demonstrate that our estimates are obtained in a very small computation time and are accurate when compared to MC simulation. © Scripta Technica, Inc. Electron Comm Jpn Pt 1, 80(12): 28–37, 1997

Vehicle routing problem belongs to classical Combination Optimization hard problem and has been extensively studied by many researchers. Vehicle Routing Problem with Weight Coefficients and Stochastic Demands (WVRPSD) is established and a new implementation for importance sampling method to solve this model is proposed in this paper, in which the classical exponential change of measure is adopted to construct the family of importance sampling distributions, and the optimal importance sampling distribution is obtained by minimizing the variance of importance sampling estimator. Numerical experiments have been conducted and the results indicate that the method can effectively solve this problem.

This dissertation focuses on the importance sampling (IS) variance reduction technique in the Monte Carlo simulation of the performance of digital communication systems over a memoryless, additive Gaussian noise channel. The result is a unbiased estimate of the decision error with significantly smaller estimation variance than conventional Monte Carlo simulation provides. Several common digital (n, k) block codes are used to illustrate the importance sampling approach. Bounds on the variance of the error estimate are developed where the channel description is simple, but where the multiple dimension decision regions are potentially complex. Coherent detection is used as a venue for the comparison of these IS techniques with respect to increasing dimension. Bounds based on the union bound are presented for several IS techniques. An overlap stratification variance reduction technique is introduced and extended with the application of IS. A mixed IS biasing function is employed for concentrating more samples in a particular error region and is employed for several of the common lower rate ([special characters omitted] < 1) block codes. For the random phase channel the IS biasing function is developed as a 2n dimensional pdf consisting of n Rayleigh-Rician pdf pairs. The error performance of several common lower rate block codes is presented. An alternate formulation, having the potential advantage of being able to use both mean translation and variance scaling, is presented. This approach is based on a commonly used technique to numerically generate a single Rayleigh-Rician pdf pair.

The authors extend the formulation for both classical Monte Carlo (MC) and importance sampling (IS) techniques to include the bit error rate (BER) and symbol error rate (SER) of multilevel systems based on symbol by symbol observation. A new IS technique and analysis of BER and SER based on the extended formulation are presented. The authors briefly summarize the previous work on BER of binary systems. The extended formulation is then presented. The conventional IS technique for BER of binary systems is shown to be applicable in simulating BER and SER of M-ary systems, and a new improved IS technique is presented. Numerical results for all the methods are provided. >

Modified efficiency importance sampling (MEIS), an importance sampling technique which can be used to yield a significant saving of run time for the assessment of bit error rate (BER) in digital communication systems using computer simulation, is described. The concept of equivalent noise source is introduced to avoid the effect of system memory on the efficiency of importance sampling, and a nonGaussian probability density function (p.d.f.) is selected as the biased equivalent noise to further improve run time. MEIS has been used successfully in the simulation of M-ary quadrature amplitude modulation (MQAM) systems. The MEIS is shown to be effective in the simulation of systems with large memory length, and in mildly nonlinear systems. The experimental results show that MEIS is a great improvement over Monte Carlo (MC) methods and that is even more efficient than the efficient importance sampling (EIS) technique. >

We review two types of adaptive Monte Carlo methods for rare event simulations. These methods are based on importance sampling. The first approach selects importance sampling distributions by minimizing the variance of importance sampling estimator. The second approach selects importance sampling distributions by minimizing the cross entropy to the optimal importance sampling distribution. We also review the basic concepts of importance sampling in the rare event simulation context. To make the basic concepts concrete, we introduce these ideas via the study of rare events of M/M/1 queues.

In this paper, we develop the use of an adaptive Importance Sampling (IS) technique in neural network training, for applications to detection in communication systems. Some topics are reconsidered, such as modifications of the error probability objective function (Pe), optimal and suboptimal IS probability density functions (biasing density functions), and adaptive importance sampling. A genetic algorithm was used for the neural network training, having utilized an adaptive IS technique for improving Pe estimations in each iteration of the training. Also, some simulation results of the training process are included in this paper.

B-ISDN networks based on asynchronous transfer mode (ATM) are characterized by very low cell loss probability. This results in an enormous computational burden when determining the system's performance via Monte Carlo simulation. To improve the computational efficiency different methods (for example Restart and GEVT) have been proposed in literature to evaluate the occurrence of rare events. We show how importance sampling (IS) allows one to obtain useful results in this field. While IS techniques have been largely exploited in the evaluation of bit error rate in digital communication systems and of false alarm rate in radar devices very little has been done to develop efficient IS-based schemes to simulate a queueing system fed with self-similar traffic. With this work we show how IS may be applied to simulate a queue fed both with fractional Gaussian noise (FGN) and with synthetic traces with characteristics close to the ones of real traffic, resulting from videoconferencing sessions.

This paper surveys efficient techniques for estimating, via simulation, the probabilities of certain rare events in queueing and reliability models. The rare events of interest are long waiting times or buffer overflows in queueing systems, and system failure events in reliability models of highly dependable computing systems. The general approach to speeding up such simulations is to accelerate the occurrence of the rare events by using importance sampling. In importance sampling, the system is simulated using a new set of input probability distributions, and unbiased estimates are recovered by multiplying the simulation output by a likelihood ratio. Our focus is on describing asymptotically optimal importance sampling techniques. Using asymptotically optimal importance sampling, the number of samples required to get accurate estimates grows slowly compared to the rate at which the probability of the rare event approaches zero. In practice, this means that run lengths can be reduced by many orders of magnitude, compared to standard simulation. In certain cases, asymptotically optimal importance sampling results in estimates having bounded relative error. With bounded relative error, only a fixed number of samples are required to get accurate estimates, no matter how rare the event of interest is. The queueing systems studied include simple queues (e.g., GI/GI/1), Jackson networks, discrete time queues with multiple autocorrelated arrival processes that arise in the analysis of Asynchronous Transfer Mode communications switches, and tree structured networks of such switches. Both Markovian and non-Markovian reliability models are treated.

In this paper, the Monte Carlo simulation is applied to reliability assessment of frame structures. Importance sampling technique is used to reduce the sample size that is necessarily large for small failure probability problems, and Neumann expansion method is adopted to decrease the computational effort in finite element analysis. Based on the analyses of numerical results, it is discussed how to select and check importance sampling distributions. Numerical examples show that the Monte Carlo simulation combined with importance sampling and Neumann expansion is much more advantageous than direct Monte Carlo simulation.

Rare events are an important limitation of discrete-event simulation in some areas of applications. The existing rare-event simulation (RES) techniques, such as importance sampling (IS), splitting, and RESTART, have been used for RES so far. However, these techniques not only have some limitations, but also are not general or easy to use. In this paper, the partition of the region (PoR) method is used as a variance reduction technique in a new RES technique for usage with a wide range of modeling and simulation languages. Three variants of the proposed technique will be introduced: simple PoR, recursive PoR, and advanced PoR. Simple and recursive PoR methods are powerful techniques that have some important features such as bounded relative error; however, they are restricted to some specific models. Advanced PoR is a more general technique that integrates the PoR method and the IS technique. To evaluate the efficiency of the proposed technique and its variants, the comparative simulation results of some illustrative examples using stochastic activity networks will be presented.

Critical events that occur rarely in biological processes are of great importance, but are challenging to study using Monte Carlo simulation. By introducing biases to reaction selection and reaction rates, weighted stochastic simulation algorithms based on importance sampling allow rare events to be sampled more effectively. However, existing methods do not address the important issue of barrier crossing, which often arises from multistable networks and systems with complex probability landscape. In addition, the proliferation of parameters and the associated computing cost pose significant problems. Here we introduce a general theoretical framework for obtaining optimized biases in sampling individual reactions for estimating probabilities of rare events. We further describe a practical algorithm called adaptively biased sequential importance sampling (ABSIS) method for efficient probability estimation. By adopting a look-ahead strategy and by enumerating short paths from the current state, we estimate the reaction-specific and state-specific forward and backward moving probabilities of the system, which are then used to bias reaction selections. The ABSIS algorithm can automatically detect barrier-crossing regions, and can adjust bias adaptively at different steps of the sampling process, with bias determined by the outcome of exhaustively generated short paths. In addition, there are only two bias parameters to be determined, regardless of the number of the reactions and the complexity of the network. We have applied the ABSIS method to four biochemical networks: the birth-death process, the reversible isomerization, the bistable Schlögl model, and the enzymatic futile cycle model. For comparison, we have also applied the finite buffer discrete chemical master equation (dCME) method recently developed to obtain exact numerical solutions of the underlying discrete chemical master equations of these problems. This allows us to assess sampling results objectively by comparing simulation results with true answers. Overall, ABSIS can accurately and efficiently estimate rare event probabilities for all examples, often with smaller variance than other importance sampling algorithms. The ABSIS method is general and can be applied to study rare events of other stochastic networks with complex probability landscape.

Let X 1, X 2, · ·· be independent and identically distributed random variables such that ΕΧ 1 < 0 and P (X 1 ≥ 0) ≥ 0. Fix M ≥ 0 and let T = inf {n: X 1 + X 2 + · ·· + Xn ≥ M} (T = +∞, if for every n = 1,2, ···). In this paper we consider the estimation of the level-crossing probabilities P (T <∞) and , by using Monte Carlo simulation and especially importance sampling techniques. When using importance sampling, precision and efficiency of the estimation depend crucially on the choice of the simulation distribution. For this choice we introduce a new criterion which is of the type of large deviations theory; consequently, the basic large deviations theory is the main mathematical tool of this paper. We allow a wide class of possible simulation distributions and, considering the case that M →∞, we prove asymptotic optimality results for the simulation of the probabilities P (T <∞) and . The paper ends with an example.