Henrik Hult

MULTIVARIATE EXTREMES, AGGREGATION AND DEPENDENCE IN ELLIPTICAL DISTRIBUTIONS

In this paper, we clarify dependence properties of elliptical distributions by deriving general but explicit formulae for the coefficients of upper and lower tail dependence and spectral measures with respect to different norms. We show that an elliptically distributed random vector is regularly varying if and only if the bivariate marginal distributions have tail dependence. Furthermore, the tail dependence coefficients are fully determined by the tail index of the random vector (or equivalently of its components) and the linear correlation coefficient. Whereas Kendalls tau is invariant in the class of elliptical distributions with continuous marginals and a fixed dispersion matrix, we show that this is not true for Spearmans rho. We also show that sums of elliptically distributed random vectors with the same dispersion matrix (up to a positive constant factor) remain elliptical if they are dependent only through their radial parts.

REGULAR VARIATION FOR MEASURES ON METRIC SPACES

The foundations of regular variation for Borel measures on a com- plete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a mapping theorem and a characterization of relative compactness are derived. Regu- lar variation is defined in this general setting and several statements that are equivalent to this definition are presented. This extends the notion of regular variation for Borel measures on the Euclidean space R d to more general metric spaces. Some examples, including regular variation for Borel measures on R d , the space of continuous functions C and the Skorohod space D ,a re provided.

Functional large deviations for multivariate regularly varying random walks

We extend classical results by A. V. Nagaev [Izv Akad. Nauk UzSSR Ser Fiz.-Mat. Nauk 6 (1969) 17-22, Theory Probab. Appl. 14 (1969) 51-64, 193-208] on large deviations for sums of i.i.d. regularly varying random variables to partial sum processes of i.i.d. regularly varying vectors. The results are stated in terms of a heavy-tailed large deviation principle on the space of cAdlAg functions. We illustrate how these results can be applied to functionals of the partial sum process, including ruin probabilities for multivariate random walks and long strange se-ments. These results make precise the idea of heavy-tailed large deviation heuristics: in an asymptotic sense, only the largest step contributes to the extremal behavior of a multivariate random walk.

Extremal behavior of stochastic integrals driven by regularly varying Levy processes

We study the extremal behavior of a stochastic integral driven by a multivariate Levy process that is regularly varying with index alpha > 0. For predictable integrands with a finite (alpha + delta)-moment, for some delta > 0, we show that the extremal behavior of the stochastic integral is due to one big jump of the driving Levy process and we determine its limit measure associated with regular variation on the space of cadlag functions.

Approximating some volterra type stochastic integrals with applications to parameter estimation

We consider Volterra type processes which are Gaussian processes admitting representation as a Volterra type stochastic integral with respect to the standard Brownian motion, for instance the fractional Brownian motion. Gaussian processes can be represented as a limit of a sequence of processes in the associated reproducing kernel Hilbert space and as a special case of this representation, we derive Karhunen-Loeve expansions for Volterra type processes. In particular, a wavelet decomposition for the fractional Brownian motion is obtained. We also consider a Skorohod type stochastic integral with respect to a Volterra type process and using the Karhunen-Loeve expansions we show how it can be approximated. Finally, we apply the results to estimation of drift parameters in stochastic models driven by Volterra type processes using a Girsanov transformation and we prove consistency, the rate of convergence and asymptotic normality of the derived maximum likelihood estimators.

On regular variation for infinitely divisible random vectors and additive processes

We study the tail behavior of regularly varying infinitely divisible random vectors and additive processes, i.e. stochastic processes with independent but not necessarily stationary increments. We show that the distribution of an infinitely divisible random vector is tail equivalent to its Lévy measure and we study the asymptotic decay of the probability for an additive process to hit sets far away from the origin. The results are extensions of known univariate results to the multivariate setting; we exemplify some of the difficulties that arise in the multivariate case.

Ruin probabilities under general investments and heavy-tailed claims

In this paper, the asymptotic decay of finite time ruin probabilities is studied. An insurance company is considered that faces heavy-tailed claims and makes investments in risky assets whose prices evolve according to quite general semimartingales. In this setting, the ruin problem corresponds to determining hitting probabilities for the solution to a randomly perturbed stochastic integral equation. A large deviation result for the hitting probabilities is derived that holds uniformly over a family of semimartingales. This result gives the asymptotic decay of finite time ruin probabilities under sufficiently conservative investment strategies, including ruin-minimizing strategies. In particular, as long as the insurance company invests sufficiently conservatively, the investment strategy has only a moderate impact on the asymptotics of the ruin probability.

On importance sampling with mixtures for random walks with heavy tails

State-dependent importance sampling algorithms based on mixtures are considered. The algorithms are designed to compute tail probabilities of a heavy-tailed random walk. The increments of the random walk are assumed to have a regularly varying distribution. Sufficient conditions for obtaining bounded relative error are presented for rather general mixture algorithms. Two new examples, called the generalized Pareto mixture and the scaling mixture, are introduced. Both examples have good asymptotic properties and, in contrast to some of the existing algorithms, they are very easy to implement. Their performance is illustrated by numerical experiments. Finally, it is proved that mixture algorithms of this kind can be designed to have vanishing relative error.

Large deviations for weighted empirical measures arising in importance sampling

In this paper the efficiency of an importance sampling algorithm is studied by means of large deviations for the associated weighted empirical measure. The main result, stated as a Laplace principle for these weighted empirical measures, can be viewed as an extension of Sanov’s theorem. The main theorem is used to quantify the performance of an importance sampling algorithm over a collection of subsets of a given target set as well as quantile estimates. The analysis yields an estimate of the sample size needed to reach a desired precision and of the reduction in cost compared to standard Monte Carlo.

Rare-event simulation for stochastic recurrence equations with heavy-tailed innovations

In this article, rare-event simulation for stochastic recurrence equations of the form <i>X</i>_<i>n</i>+1=<i>A</i>_<i>n</i>+1<i>X</i>_<i>n</i>+<i>B</i>_<i>n</i>+1, <i>X</i>₀=0 is studied, where {<i>A</i>_n;<i>n</i>≥ 1} and {<i>B</i>_n;<i>n</i>≥ 1} are independent sequences consisting of independent and identically distributed real-valued random variables. It is assumed that the tail of the distribution of <i>B</i>₁ is regularly varying, whereas the distribution of <i>A</i>₁ has a suitably light tail. The problem of efficient estimation, via simulation, of quantities such as <i>P</i>{<i>X</i>_<i>n</i>>b} and <i>P</i>{sup_{<i>k</i>≤<i>n</i>}<i>X</i>_k > b} for large <i>b</i> and <i>n</i> is studied. Importance sampling strategies are investigated that provide unbiased estimators with bounded relative error as <i>b</i> and <i>n</i> tend to infinity.