We introduce a definition of long range dependence of random processes and fields on an (unbounded) index space $T\subseteq \mathbb{R}^d$ in terms of integrability of the covariance of indicators that a random function exceeds any given level. This definition is specifically designed to cover the case of random functions with infinite variance. We show the value of this new definition and its connection to limit theorems via some examples including subordinated Gaussian as well as random volatility fields and time series.

Catastrophe insurance markets fail to provide sufficient protections against natural catastrophes, whereas they have the capacity to absorb the losses. In this paper, we assume the catastrophic risks are dependent and extremely heavy-tailed, and insurers have limited liability to cover losses up to a certain amount. We provide a comprehensive study to show that the diversification in the catastrophe insurance markets can be transited from suboptimal to preferred by increasing the number of insurers in the market. This highlights the importance of coordination among insurers and the government intervention in encouraging insurers to participate in the catastrophe insurance market to exploit risk sharing. Simulation studies are provided to illuminate the key findings of our results.

We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable the carrying over of fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach in modelling a real-world heavy-tailed fire insurance dataset.

Recent progress in microdissection and in DNA sequencing has facilitated the subsampling of multi-focal cancers in organs such as the liver in several hundred spots, helping to determine the pattern of mutations in each of these spots. This has led to the construction of genealogies of the primary, secondary, tertiary, and so forth, foci of the tumor. These studies have led to diverse conclusions concerning the Darwinian (selective) or neutral evolution in cancer. Mathematical models of the development of multi-focal tumors have been devised to support these claims. We offer a model for the development of a multi-focal tumor: it is a mathematically rigorous refinement of a model of Ling et al. (2015). Guided by numerical studies and simulations, we show that the rigorous model, in the form of an infinite-type branching process, displays distributions of tumor size which have heavy tails and moments that become infinite in finite time. To demonstrate these points, we obtain bounds on the tails of the distributions of the process and an infinite series expansion for the first moments. In addition to its inherent mathematical interest, the model is corroborated by recent literature on apparent super-exponential growth in cancer metastases.

Validation of risk models is required by regulators and demanded by management and shareholders. Those models rely in practice heavily on Monte Carlo (MC) simulations. Given their complexity, the convergence of the MC algorithm is difficult to prove mathematically. To circumvent this problem and nevertheless explore the conditions of convergence, we suggest an analytical approach. Considering standard models, we compute, via mixing techniques, closed form formulas for risk measures as Value-at-Risk (VaR) VaR or Tail Value-at-Risk (TVaR) TVaR on a portfolio of risks, and consequently for the associated diversification benefit. The numerical convergence of MC simulations of those various quantities is then tested against their analytical evaluations. The speed of convergence appears to depend on the fatness of the tail of the marginal distributions; the higher the tail index, the faster the convergence. We also explore the behaviour of the diversification benefit with various dependence structures and marginals (heavy and light tails). As expected, it varies heavily with the type of dependence between aggregated risks. The diversification benefit is also studied as a function of the risk measure, VaR or TVaR.

We study how to sample paths of a random walk up to the first time it crosses a fixed barrier, in the setting where the step sizes are independent and identically distributed with negative mean and have a regularly varying right tail. We introduce a desirable property for a change of measure to be suitable for exact simulation. We study whether the change of measure of Blanchet and Glynn (2008) satisfies this property and show that it does so if and only if the tail index α of the right tail lies in the interval (1, 3/2).

Taylor's law (TL) originated as an empirical pattern in ecology. In many sets of samples of population density, the variance of each sample was approximately proportional to a power of the mean of that sample. In a family of nonnegative random variables, TL asserts that the population variance is proportional to a power of the population mean. TL, sometimes called fluctuation scaling, holds widely in physics, ecology, finance, demography, epidemiology, and other sciences, and characterizes many classical probability distributions and stochastic processes such as branching processes and birth-and-death processes. We demonstrate analytically for the first time that a version of TL holds for a class of distributions with infinite mean. These distributions, a subset of stable laws, and the associated TL differ qualitatively from those of light-tailed distributions. Our results employ and contribute to the methodology of Albrecher and Teugels (2006) and Albrecher et al. (2010). This work opens a new domain of investigation for generalizations of TL.

We consider a resource allocation model to analyze investment strategies for financial services firms in order to minimize their operational risk losses. A firm has to decide how much to invest in human resources and in infrastructure (information technology). The operational risk losses are a function of the activity level of the firm, of the amounts invested in personnel and in infrastructure, and of interaction effects between the amounts invested in personnel and infrastructure. We first consider a deterministic setting and show certain monotonicity properties of the optimal investments assuming general loss functions that are convex. We find that because of the interaction effects “economies of scale" may not hold in our setting, in contrast to a typical manufacturing environment. We then consider a general polynomial loss function in a stochastic setting with the number of transactions at the firm being a random variable. We characterize the asymptotic behaviors of the optimal investments in both heavy and light trading environments. We show that when the market is very liquid, that is, it is subject to heavy transaction volumes, it is optimal for a financial firm that is highly risk sensitive to use a balanced investment strategy. Both a heavier right tail of the distribution of transaction volume and a firm's risk sensitivity necessitate larger investments; in a heavy trading environment these two factors reinforce one another. However, in a light trading environment with the transaction volume having a heavy left tail the investment will be independent of the firm's sensitivity to risk.

A large deviations principle is established for the joint law of the empirical measure and the flow measure of a Markov renewal process on a finite graph. We do not assume any bound on the arrival times, allowing heavy-tailed distributions. In particular, the rate function is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behaviour highly different from what one may guess with a heuristic Donsker‒Varadhan analysis of the problem.

We study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

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.

An asymptotic model for the extreme behavior of certain Markov chains is the ‘tail chain’. Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics, such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and nonextreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.

In this paper we investigate the asymptotic behaviors of the finite- and infinite-time ruin probabilities for a Poisson risk model with stochastic investment returns which constitute a general adapted càdlàg process and heavy-tailed claim sizes which are bivariate upper tail independent. The results of this paper show that the asymptotic ruin probabilities are dominated by the extreme of insurance risk but not by that of investment risk. As applications of the results, we discuss four special cases when the investment returns are determined by a fractional Brownian motion, an integrated Vasicek model, an integrated Cox–Ingersoll–Ross model, and the Heston model.

We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate, and which captures some relevant stylized facts of financial assets, including scaling properties. In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation, and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average, with remarkable accuracy.

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

Let {Xi} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C1, C2,…) be a nonnegative random vector independent of the {Xi} with N∈ℕ∪ {∞}. We study the weighted random sum SN=∑{i=1}NCiXi, and its maximum, MN=sup{1≤kN+1∑i=1kCiXi. This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(MN > x)∼ P(SN > x)∼ E[∑i=1NF̄(x/Ci)] as x→∞. When E[X1]>0 and the distribution of ZN=∑ i=1NCi is also intermediate regularly varying, we obtain the asymptotics P(MN > x)∼ P(SN > x)∼ E[∑i=1NF̄}(x/Ci)] +P(ZN > x/E[X1]). For completeness, when the distribution of ZN is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(MN > x) ∼ P(SN > x)∼ P(ZN > x / E[X1] hold.

Let {Xt, t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θt, t ≥ 1} be a sequence of positive random variables independent of the sequence {Xt, t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X(∞) = ∑t=1∞ΘtXt+ (where X+ = max{0, X}) and max1≤k<∞∑t=1kΘtXt, and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X1, X2,… as independent and identically distributed positive random variables. If X1 has a regularly varying tail of index -α, where α > 0, and if {Θt, t ≥ 1} is a positive sequence of random variables independent of {Xt}, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θt, t ≥ 1}, X(∞) = ∑t=1∞ΘtXt converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X(∞) has a regularly varying tail then does X1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑t=1∞Θt along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.

In this paper we consider the probabilities of finite- and infinite-time absolute ruins in the renewal risk model with constant premium rate and constant force of interest. In the particular case of the compound Poisson model, explicit asymptotic expressions for the finite- and infinite-time absolute ruin probabilities are given. For the general renewal risk model, we present an asymptotic expression for the infinite-time absolute ruin probability. Conditional distributions of Poisson processes and probabilistic techniques regarding randomly weighted sums are employed in the course of this study.

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.

We consider a nearly unstable, or near unit root, AR(1) process with regularly varying innovations. Two different approximations for the stationary distribution of such processes exist: a Gaussian approximation arising from the nearly unstable nature of the process and a heavy-tail approximation related to the tail asymptotics of the innovations. We combine these two approximations to obtain a new uniform approximation that is valid on the entire real line. As a corollary, we obtain a precise description of the regions where each of the Gaussian and heavy-tail approximations should be used.