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For fractional Brownian motion with Hurst parameter H the Berman constant is defined. In this paper we consider a general random field (rf) Z that is a spectral rf of some stationary max-stable rf X and derive the properties of the corresponding Berman functions. In particular, we show that Berman functions can be approximated by the corresponding discrete ones and derive interesting representations of those functions which are of interest for Monte Carlo simulations, which are presented in this article.

For a non-negative separable random field $Z(t), t\in \mathbb{R}^d$ satisfying some mild assumptions we show that \begin{eqnarray*} H_Z^\delta = \lim_{T\to\infty} \frac{1}{T^d} E \{\sup_{ t\in [0,T]^d \cap \delta \mathbb{Z}^d } Z(t) \} <\infty \end{eqnarray*} for $\delta \ge 0$ where $0 \mathbb{Z}^d := \mathbb{R}^d$ and prove that $H_Z^0$ can be approximated by $H_Z^\delta$ if $\delta$ tends to 0. These results extend the classical findings for the Pickands constants $H_{Z}^\delta$, defined for $Z(t)= \exp\left( \sqrt{ 2} B_\alpha (t)- |t|^{2\alpha }\right), t\in \mathbb{R}$ with $B_\alpha$ a standard fractional Brownian motion with Hurst parameter $\alpha \in (0,1]$. The continuity of $H_{Z}^\delta$ at $\delta=0$ is additionally shown for two particular extensions of Pickands constants., Comment: 20 pages

This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum o f random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes., Comment: 31 pages

The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite-time horizon. This is not the case for the simultaneous ruin probability in two-dimensional Brownian risk model. Resorting on asymptotic theory, we derive in this contribution approximations of both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity. Given the interest in proportional reinsurance, we consider in some details the case where the correlation is 1. This model is tractable allowing for explicit formulas for the simultaneous ruin probability for linearly dependent spectrally positive L\'evy processes. Examples include perturbed Brownian and gamma L\'evy processes., Comment: 16 pages

We derive exact tail asymptotics of sojourn time above the level $u\geq 0$ $$ \mathbb{P}\left(v(u)\int_0^T \mathbb{I}(X(t)-ct>u)d t>x\right), \quad x\geq 0 $$ as $u\to\infty$, where $X$ is a Gaussian process with continuous sample paths, $c>0$, $v(u)$ is a positive function of $u$ and $T\in (0,\infty]$. Additionally, we analyze asymptotic distributional properties of $$\tau_u(x):=\inf\left\{t\geq 0: v(u) \int_0^t \mathbb{I}(X(s)-cs>u)d s>x\right\}, $$ as $u\to\infty$, $x\geq 0$, where $\inf\emptyset=\infty$. The findings of this contribution are illustrated by a detailed analysis of a class of Gaussian processes with stationary increments and a family of self-similar processes., Comment: 30 pages

In this note we find a formula for the supremum distribution of spectrally positive or negative L\'evy processes with a broken linear drift. This gives formulas for ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specified proportions. As an example we consider gamma L\'evy process, $\alpha$-stable L\'evy process and Brownian motion. Moreover we obtain identities for Laplace transform of the distribution for the supremum of L\'evy processes with randomly broken drift and on random intervals.

We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes $X$, in both continuous- and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time case, we complement the investigations of [1,2] for non-stationary $X$. A by-product of our investigation is a new representation of Pickands constant which is important for Monte-Carlo simulations and yields a sharp lower bound for Pickands constant., Comment: 26 pages

We quantify the bullwhip effect (which measures how the variance in replenishment orders is amplified as the orders move up the supply chain) when random demands and random lead times are estimated using the industrially popular moving average forecasting method. We assume that the lead times constitute a sequence of independent identically distributed random variables and correlated demands are described by a first order autoregressive process. We obtain an expression that reveals the impact of demand and lead time forecasting on the bullwhip effect. We draw a number of conclusions on the behavior of the bullwhip effect with respect to the demand auto-correlation and the number of past lead times and demands used in the forecasts. Furthermore we find the maxima and minima of the bullwhip measure as a function of the demand auto-correlation.

In this article we want to review the research state on the bullwhip effect in supply chains with stochastic lead times and give a contribution to quantifying the bullwhip effect. We analyze the models quantifying the bullwhip effect in supply chains with stochastic lead times and find advantages and disadvantages of their approaches to the bullwhip problem. Using real data we confirm that real lead times are stochastic and can be modeled by a sequence of independent identically distributed random variables. Moreover we modify a model where stochastic lead times and lead time demand forecasting are considered and give an analytical expression for the bullwhip effect measure which indicates that the distribution of a lead time and the delay parameter of the lead time demand prediction are the main factors of the bullwhip phenomenon. Moreover we analyze a recent paper of Michna and Nielsen adding simulation results.

In this article we derive formulas for the probability $P(\sup_{t\leq T} X(t)>u)$ $T>0$ and $P(\sup_{t<\infty} X(t)>u)$ where $X$ is a spectrally positive L\'evy process with infinite variation. The formulas are generalizations of the well-known Tak\'acs formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of $\inf_{t\leq T} Y(t)$ and $Y(T)$ where $Y$ is a spectrally negative L\'evy process.

In this article we quantify the bullwhip effect (the variance amplification in replenishment orders) when demands and lead times are predicted in a simple two-stage supply chain with one supplier and one retailer. In recent research the impact of stochastic order lead time on the bullwhip effect is investigated, but the effect of needing to predict / estimate the lead time is not considered in the supply chain models. Under uncertainty conditions it is necessary to estimate the lead time for a member of the supply chain to place an order. We find a new cause of the bullwhip effect in the form of lead time forecasting and we give an exact form of the bullwhip effect measure (the ratio of variances) when demands and lead times are predicted by moving averages. In the bullwhip effect measure we discover two terms amplifying the effect which are the result of lead time estimation

In this paper we consider storage and inventory systems. Our aim is to apply and review main results of the fluctuation theory of stochastic processes in the context of storage and inventory modeling. We describe systems where the inflow is due to a L\'evy process and the outflow is linear and conversely systems where the inflow is linear and the outflow is due to a L\'evy process. For such systems we investigate the process of a storage (inventory) level. We give formulas for the probability that the storage level exceeds a certain value. This probability is crucial in the management of storage and inventory systems.

In this article we get simple explicit formulas for $\Exp\sup_{s\leq t}X(s)$ where $X$ is a spectrally positive or negative L\'evy process with infinite variation. As a consequence we derive a generalization of the well-known formula for the supremum distribution of Wiener process that is we obtain $\Prob(\sup_{s\leq t}Z_{\alpha}(s)\geq u)=\alpha \Prob(Z_{\alpha}(t)\geq u)$ for $u\geq 0$ where $Z_{\alpha}$ is a spectrally negative L\'evy process with $1<\alpha\leq 2$ which also stems from Kendall's identity for the first crossing time. Our proof uses a formula for the supremum distribution of a spectrally positive L\'evy process which follows easily from the elementary Seals formula.

In this article we derive formula for probability $\Prob(\sup_{t\leq T} (X(t)-ct)>u)$ where $X=\{X(t)\}$ is a spectrally positive L\'evy process and $c\in\RL$. As an example we investigate the inverse Gaussian L\'evy process.

In this article we present Pickands theorem and his double sum method. We follow Piterbarg's proof of this theorem. Since his proof relies on general lemmas we present a complete proof of Pickands theorem using Borell inequality and Slepian lemma. The original Pickands proof is rather complicated and is mixed with upcrossing probabilities for stationary Gaussian processes. We give a lower bound for Pickands constant.