This paper focuses on the evaluation of the probability that both components of a bivariate stochastic process ever simultaneously exceed some large level; a leading example is that of two Markov fluid queues driven by the same background process ever reaching the set (u, ∞)×(u, ∞), for u>0. Exact analysis being prohibitive, we resort to asymptotic techniques and efficient simulation, focusing on large values of u. The first contribution concerns various expressions for the decay rate of the probability of interest, which are valid under Gärtner–Ellis-type conditions. The second contribution is an importance-sampling-based rare-event simulation technique for the bivariate Markov modulated fluid model, which is capable of asymptotically efficiently estimating the probability of interest; the efficiency of this procedure is assessed in a series of numerical experiments.

In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

We study asymptotically optimal simulation algorithms for approximating the tail probability of P(eX1+⋯+ eXd>u) as u→∞. The first algorithm proposed is based on conditional Monte Carlo and assumes that (X1,…,Xd) has an elliptical distribution with very mild assumptions on the radial component. This algorithm is applicable to a large class of models in finance, as we demonstrate with examples. In addition, we propose an importance sampling algorithm for an arbitrary dependence structure that is shown to be asymptotically optimal under mild assumptions on the marginal distributions and, basically, that we can simulate efficiently (X1,…,Xd|Xj >b) for large b. Extensions that allow us to handle portfolios of financial options are also discussed.

The variance minimization (VM) and cross-entropy (CE) methods are two versatile adaptive importance sampling procedures that have been successfully applied to a wide variety of difficult rare-event estimation problems. We compare these two methods via various examples where the optimal VM and CE importance densities can be obtained analytically. We find that in the cases studied both VM and CE methods prescribe the same importance sampling parameters, suggesting that the criterion of minimizing the CE distance is very close, if not asymptotically identical, to minimizing the variance of the associated importance sampling estimator.

We consider the problem of estimating the probability that the maximum of a Gaussian process with negative mean and indexed by positive integers reaches a high level, say b. In great generality such a probability converges to 0 exponentially fast in a power of b. Under mild assumptions on the marginal distributions of the process and no assumption on the correlation structure, we develop an importance sampling procedure, called the target bridge sampler (TBS), which takes a polynomial (in b) number of function evaluations to achieve a small relative error. The procedure also yields samples of the underlying process conditioned on hitting b in finite time. In addition, we apply our method to the problem of estimating the tail of the maximum of a superposition of a large number, n, of independent Gaussian sources. In this situation TBS achieves a prescribed relative error with a bounded number of function evaluations as n ↗ ∞. A remarkable feature of TBS is that it is not based on exponential changes of measure. Our numerical experiments validate the performance indicated by our theoretical findings.

We consider the problem of efficient estimation via simulation of first passage time probabilities for a multidimensional random walk with heavy-tailed increments. In addition to being a natural generalization to the problem of computing ruin probabilities in insurance - in which the focus is the maximum of a one-dimensional random walk with negative drift - this problem captures important features of large deviations for multidimensional heavy-tailed processes (such as the role played by the mean of the process in connection to the location of the target set). We develop a state-dependent importance sampling estimator for this class of multidimensional problems. Then, using techniques based on Lyapunov inequalities, we argue that our estimator is strongly efficient in the sense that the relative mean squared error of our estimator can be made arbitrarily small by increasing the number of replications, uniformly as the probability of interest approaches 0.

The standard Markov chain Monte Carlo method of estimating an expected value is to generate a Markov chain which converges to the target distribution and then compute correlated sample averages. In many applications the quantity of interest θ is represented as a product of expected values, θ = µ1 ⋯ µk, and a natural estimator is a product of averages. To increase the confidence level, we can compute a median of independent runs. The goal of this paper is to analyze such an estimator , i.e. an estimator which is a ‘median of products of averages’ (MPA). Sufficient conditions are given for to have fixed relative precision at a given level of confidence, that is, to satisfy . Our main tool is a new bound on the mean-square error, valid also for nonreversible Markov chains on a finite state space.

Consider a sequence (Xk: k ≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (Sn: n ≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail of Sn in a large deviation regime as n ↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.