Efficient rare-event simulation for the maximum of heavy-tailed random walks

Let $(X_n:n\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean. Set $S_0=0$ and define $S_n=X_1+... +X_n$. We propose an importance sampling algorithm to estimate the tail of $M=\max \{S_n:n\geq 0\}$ that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmund's (1979) algorithm. The rigorous analysis of efficiency requires new Lyapunov-type inequalities that can be useful in the study of more general importance sampling algorithms.Comment: Published in at http://dx.doi.org/10.1214/07-AAP485 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Efficient Simulation and Conditional Functional Limit Theorems for Ruinous Heavy-tailed Random Walks

The contribution of this paper is to introduce change of measure based techniques for the rare-event analysis of heavy-tailed stochastic processes. Our changes-of-measure are parameterized by a family of distributions admitting a mixture form. We exploit our methodology to achieve two types of results. First, we construct Monte Carlo estimators that are strongly efficient (i.e. have bounded relative mean squared error as the event of interest becomes rare). These estimators are used to estimate both rare-event probabilities of interest and associated conditional expectations. We emphasize that our techniques allow us to control the expected termination time of the Monte Carlo algorithm even if the conditional expected stopping time (under the original distribution) given the event of interest is infinity -- a situation that sometimes occurs in heavy-tailed settings. Second, the mixture family serves as a good approximation (in total variation) of the conditional distribution of the whole process given the rare event of interest. The convenient form of the mixture family allows us to obtain, as a corollary, functional conditional central limit theorems that extend classical results in the literature. We illustrate our methodology in the context of the ruin probability $P(\sup_n S_n >b)$, where $S_n$ is a random walk with heavy-tailed increments that have negative drift. Our techniques are based on the use of Lyapunov inequalities for variance control and termination time. The conditional limit theorems combine the application of Lyapunov bounds with coupling arguments

Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks

We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in DeanDup09 that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffice to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n^{2{\beta}+1}) function evaluations suffice to achieve a given relative precision, where {\beta} is the number of bottleneck stations in the network. This is the first rigorous analysis that allows to favorably compare splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with O(n^{d}) variables.Comment: 23 page

Efficient Rare-event Simulation for Perpetuities

We consider perpetuities of the form D = B_1 exp(Y_1) + B_2 exp(Y_1+Y_2) + ... where the Y_j's and B_j's might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the Y_j's satisfy the so-called Cramer condition with associated root theta_{ast} in (0,infty) and that the tails of the B_j's are appropriately behaved so that D is regularly varying with index theta_{ast}. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the Y_j's according to theta_{ast} fails to provide an efficient estimator (in the sense of appropriately controlling the relative mean squared error as the tail probability of interest gets smaller). Then, we construct estimators based on state-dependent importance sampling that are rigorously shown to be efficient