In this paper we address the problem of rare-event simulation for heavy-tailed Lévy processes with infinite activities. We propose a strongly efficient importance sampling algorithm that builds upon the sample path large deviations for heavy-tailed Lévy processes, stick-breaking approximation of extrema of Lévy processes, and the randomized debiasing Monte Carlo scheme. The proposed importance sampling algorithm can be applied to a broad class of Lévy processes and exhibits significant improvements in efficiency when compared to crude Monte-Carlo method in our numerical experiments.
We develop a framework for regularly varying measures on complete separable metric spaces $\mathbb{S}$ with a closed cone $\mathbb{C}$ removed, extending material in Hult & Lindskog (2006), Das, Mitra & Resnick (2013). Our framework provides a flexible way to consider hidden regular variation and allows simultaneous regular variation properties to exist at different scales and provides potential for more accurate estimation of probabilities of risk regions. We apply our framework to iid random variables in $\mathbb{R}_+^\infty$ with marginal distributions having regularly varying tails and to …
This paper addresses heavy-tailed large deviation estimates for the distribution tail of functionals of a class of spectrally one-sided Lévy process. Our contribution is to show that these estimates remain valid in a near-critical regime. This complements recent similar results that have been obtained for the all-time supremum of such processes. Specifically, we consider local asymptotics of the all-time supremum, the supremum of the process until exiting $[0,\infty)$, the maximum jump until that time, and the time it takes until exiting $[0,\infty)$. The proofs rely, among other things, on pr…
We propose a class of strongly efficient rare event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on the heavy-tailed sample path large deviations result recently established in Rhee, Blanchet, and Zwart (2016). The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are "universal" in the…
Let $X$ be a Lévy process with regularly varying Lévy measure $ν$. We obtain sample-path large deviations for scaled processes $\bar X_n(t) \triangleq X(nt)/n$ and obtain a similar result for random walks. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but…
We study the extremal behavior of a stochastic integral driven by a multivariate Lévy process that is regularly varying with index $α>0$. For predictable integrands with a finite $(α+δ)$-moment, for some $δ>0$, we show that the extremal behavior of the stochastic integral is due to one big jump of the driving Lévy process and we determine its limit measure associated with regular variation on the space of càdlàg functions.
We consider subordinators $X_α=(X_α(t))_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator $(S_α(t))_{t\ge 0}$ (where $α\in(0,1)$); thus, with the property that $\overlineΠ_α$, the tail function of the canonical measure of $X_α$, is regularly varying of index $-α\in (-1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $α=0$, when $\overlineΠ_α$ is slowly varying at 0. When $α\in(0,1)$, we show that $(t \overlineΠ_α(X_α(t)))^{-1}$ converges in distribution, as $t\downarrow 0$, to the random variable $(S_α(1))^α$. This latter random variable, as a function of $α$, conv…
We consider the sample average of a centered random walk in $\mathbb{R}^d$ with regularly varying step size distribution. For the first exit time from a compact convex set $A$ not containing the origin, we show that its tail is of lognormal type. Moreover, we show that the typical way for a large exit time to occur is by having a number of jumps growing logarithmically in the scaling parameter.
For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to …
We study large deviation probabilities for a sum of dependent random variables from a heavy-tailed factor model, assuming that the components are regularly varying. We identify conditions where both the factor and the idiosyncratic terms contribute to the behaviour of the tail-probability of the sum. A simple conditional Monte Carlo algorithm is also provided together with a comparison between the simulations and the large deviation approximation. We also study large deviation probabilities for stochastic processes with factor structure. The processes involved are assumed to be Levy processes …
In this paper, we study transition density functions for pure jump unimodal Lévy processes killed upon leaving an open set $D$. Under some mild assumptions on the Lévy density, we establish two-sided Dirichlet heat kernel estimates when the open set $D$ is $C^{1, 1}$. Our result covers the case that the Lévy densities of unimodal Lévy processes are regularly varying functions whose indices are equal to the Euclidean dimension. This is the first results on two-sided Dirichlet heat kernel estimates for Lévy processes such that the weak lower scaling index of the Lévy densities is not necessarily…
Various functional limit theorems for partial sum processes of strictly stationary sequences of regularly varying random variables in the space of cadlag functions $D[0,1]$ with one of the Skorohod topologies have already been obtained. The mostly used Skorohod $J_{1}$ topology is inappropriate when clustering of large values of the partial sum processes occurs. When all extremes within each cluster of high-threshold excesses do not have the same sign, Skorohod $M_{1}$ topology also becomes inappropriate. In this paper we alter the definition of the partial sum process in order to shrink all e…
In this paper we first provide several conditional limit theorems for Lévy processes with negative drift and regularly varying tail. Then we apply them to study the asymptotic behavior of expectations of some exponential functionals of heavy-tailed Lévy processes. As the key point, we observe that the asymptotics mainly depends on the sample paths with early arrival large jump. Both the polynomial decay rate and the exact expression of the limit coefficients are given. As an application, we give an exact description for the extinction speed of continuous-state branching processes in heavy-tail…
We investigate weak convergence of finite-dimensional distributions of a renewal shot noise process $(Y(t))_{t\geq 0}$ with deterministic response function $h$ and the shots occurring at the times $0 = S_0 < S_1 < S_2<\ldots$, where $(S_n)$ is a random walk with i.i.d.\ jumps. There has been an outbreak of recent activity around this topic. We are interested in one out of few cases which remained open: $h$ is regularly varying at $\infty$ of index $-1/2$ and the integral of $h^2$ is infinite. Assuming that $S_1$ has a moment of order $r>2$ we use a strong approximation argument to show that th…
Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable Lévy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable Lévy process. Due …
A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains
