International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo . June 28-30 , 2006 Extremes and ruin of Gaussian processes (original) (raw)

A limit theorem for the time of ruin in a Gaussian ruin problem

Stochastic Processes and their Applications, 2008

For certain Gaussian processes X (t) with trend −ct β and variance V 2 (t), the ruin time is analyzed where the ruin time is defined as the first time point t such that X (t) − ct β ≥ u. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications, e.g. in telecommunications where it indicates the first time of an overflow. We derive the asymptotic distribution of the ruin time as u → ∞ showing that the limiting distribution depends on the parameters β, V (t) and the correlation function of X (t).

On first and last ruin times of Gaussian processes

Statistics & Probability Letters, 2008

Considering centered Gaussian processes X (t) with a trend −ct β and variance V 2 (t), we are interested in the asymptotic distributions of the first ruin time and the last ruin time as well as their joint asymptotic distribution as the initial capital u → ∞. Our results show that the conditional distribution of the last ruin time, conditioned on ruin occurring, is a normal distribution and the conditional joint limit distribution is a difference of two standard normal distributions.

Extreme values of a portfolio of Gaussian processes and a trend

Extremes, 2005

We consider the extreme values of a portfolio of independent continuous Gaussian processes P k i¼1 w i X i ðtÞ (w i 2 R; k 2 N) which are asymptotically locally stationary, with expectations E½X i ðtÞ ¼ 0 and variances Var½X i ðtÞ ¼ d i t 2H i ðd i 2 R þ ; 0 < H i < 1Þ, and a trend Àct for some constants ; c > 0 with > H i. We derive the probability Pfsup t>0 P k i¼1 w i X i ðtÞ À ct > ug for u ! 1, which may be interpreted as ruin probability.

Extremes of Gaussian Processes with Random Variance

Electronic Journal of Probability, 2011

Let ξ(t) be a standard locally stationary Gaussian process with covariance function 1 − r(t, t + s) ∼ C(t)|s| α as s → 0, with 0 < α ≤ 2 and C(t) a positive bounded continuous function. We are interested in the exceedance probabilities of ξ(t) with a random standard deviation η(t) = η − ζt β , where η and ζ are non-negative bounded random variables. We investigate the asymptotic behavior of the extreme values of the process ξ(t)η(t) under some specific conditions which depends on the relation between α and β.

Extremes of a certain class of Gaussian processes

Stochastic Processes and their Applications, 1999

We consider the extreme values of fractional Brownian motions, self-similar Gaussian processes and more general Gaussian processes which have a trend −ct ÿ for some constants c; ÿ ¿ 0 and a variance t 2H. We derive the tail behaviour of these extremes and show that they occur mainly in the neighbourhood of the unique point t0 where the related boundary function (u + ct ÿ)=t H is minimal. We consider the case that H ¡ ÿ.

Extremes of Gaussian Processes with Maximal Variance near the Boundary Points

Methodology And Computing In Applied Probability

Let Xt, t[0Y 1, be a Gaussian process with continuous paths with mean zero and nonconstant variance. The largest values of the Gaussian process occur in the neighborhood of the points of maximum variance. If there is a unique ®xed point t 0 in the interval 0Y 1, the behavior of Pfsup t[0Y1 Xt4ug is known for u??. We investigate the case where the unique point t 0 t u depends on u and tends to the boundary. This is reasonable for a family of Gaussian processes X u t depending on u, which have for each u such a unique point t u tending to the boundary as u??. We derive the asymptotic behavior of Pfsup t [ 0Y1 Xt4ug, depending on the rate as t u tends to 0 or 1. Some applications are mentioned and the computation of a particular case is used to compare simulated probabilities with the asymptotic formula. We consider the exceedances of such a nonconstant boundary by a Ornstein-Uhlenbeck process. It shows the dif®culties to simulate such rare events, when u is large.

Tail waiting times and the extremes of stochastic processes

arXiv (Cornell University), 2015

A variety of methods have been proposed for inference about extreme dependence for multivariate or spatially-indexed stochastic processes and time series. Most of these proceed by first transforming data to some specific extreme value marginal distribution, often the unit Fréchet, then fitting a family of max-stable processes to the transformed data and exploring dependence within the framework of that model. The marginal transformation, model selection, and model fitting are all possible sources of misspecification in this approach. We propose an alternative model-free approach, based on the idea that substantial information on the strength of tail dependence and its temporal structure are encoded in the distribution of the waiting times between exceedances of high thresholds at different locations. We propose quantifying the strength of extremal dependence and assessing uncertainty by using statistics based on these waiting times. The method does not rely on any specific underlying model for the process, nor on asymptotic distribution theory. The method is illustrated by applications to climatological, financial, and electrophysiology data. To put the proposed approach within the context of the existing literature, we construct a class of spacetime-indexed stochastic processes whose waiting time distributions are available in closed form by endowing the support points in de Haan's spectral representation of maxstable processes with random birth times, velocities, and lifetimes, and applying Smith's model to these processes. We show that waiting times in this model are stochatically decreasing in mean speed, and the sample mean of the waiting times obeys a central limit theorem with a uniform convergence rate under mild conditions. This indicates that our procedure can be implemented in this setting using standard t statistics and associated hypothesis tests.

Extremes of independent Gaussian processes

Extremes, 2011

For every n ∈ N, let X 1n , . . . , Xnn be independent copies of a zero-mean Gaussian process Xn = {Xn(t), t ∈ T }. We describe all processes which can be obtained as limits, as n → ∞, of the process an(Mn − bn), where Mn(t) = max i=1,...,n X in (t) and an, bn are normalizing constants. We also provide an analogous characterization for the limits of the process anLn, where Ln(t) = min i=1,...,n |X in (t)|.

Extremes of vector-valued Gaussian processes: Exact asymptotics

Stochastic Processes and their Applications, 2015

Let {X i (t), t ≥ 0}, 1 ≤ i ≤ n be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of P ∃ t∈[0,T ] ∀ i=1,...,n X i (t) > u as u → ∞, for both locally stationary X i 's and X i 's with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants, that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell-TIS inequality, the Slepian lemma and the Pickands-Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.