arXiv subjects

Jevgenijs Ivanovs

Publications and source records attributed to Jevgenijs Ivanovs.

At least 37 records · Page 2Linked to original sources

arXiv2017

On the longest gap between power-rate arrivals

Let $L_t$ be the longest gap before time $t$ in an inhomogeneous Poisson process with rate function $λ_t$ proportional to $t^{α-1}$ for some $α\in(0,1)$. It is shown that $λ_tL_t-b_t$ has a limiting Gumbel distribution for suitable constants $b_t$ and that the distance of this longest gap from $t$ is asymptotically of the form $(t/\log t)E$ for an exponential random variable $E$. The analysis is performed via weak convergence of related point processes. Subject to a weak technical condition, the results are extended to include a slowly varying term in $λ_t$.

math.PR↗

arXiv2017

Zooming in on a Lévy process at its supremum

Let $M$ and $τ$ be the supremum and its time of a Lévy process $X$ on some finite time interval. It is shown that zooming in on $X$ at its supremum, that is, considering $((X_{τ+t\varepsilon}-M)/a_\varepsilon)_{t\in\mathbb R}$ as $\varepsilon\downarrow 0$, results in $(ξ_t)_{t\in\mathbb R}$ constructed from two independent processes having the laws of some self-similar Lévy process $\widehat X$ conditioned to stay positive and negative. This holds when $X$ is in the domain of attraction of $\widehat X$ under the zooming-in procedure as opposed to the classical zooming out of Lamperti (1962). As an application of this result we establish a limit theorem for the discretization errors in simulation of supremum and its time, which extends the result of Asmussen, Glynn and Pitman (1995) for the Brownian motion. Additionally, complete characterization of the domains of attraction when zooming in on a Lévy process at 0 is provided.

math.PR↗

arXiv2017

Robust bounds in multivariate extremes

Extreme value theory provides an asymptotically justified framework for estimation of exceedance probabilities in regions where few or no observations are available. For multivariate tail estimation, the strength of extremal dependence is crucial and it is typically modeled by a parametric family of spectral distributions. In this work we provide asymptotic bounds on exceedance probabilities that are robust against misspecification of the extremal dependence model. They arise from optimizing the statistic of interest over all dependence models within some neighborhood of the reference model. A certain relaxation of these bounds yields surprisingly simple and explicit expressions, which we propose to use in applications. We show the effectiveness of the robust approach compared to classical confidence bounds when the model is misspecified. The results are further applied to quantify the effect of model uncertainty on the Value-at-Risk of a financial portfolio.

math.PR↗

arXiv2016

Time inhomogeneity in longest gap and longest run problems

Consider an inhomogeneous Poisson process and let $D$ be the first of its epochs which is followed by a gap of size $\ell>0$. We establish a criterion for $D<\infty$ a.s., as well as for $D$ being long-tailed and short-tailed, and obtain logarithmic tail asymptotics in various cases. These results are translated into the discrete time framework of independent non-stationary Bernoulli trials where the analogue of $D$ is the waiting time for the first run of ones of length $\ell$. A main motivation comes from computer reliability, where $D+\ell$ represents the actual execution time of a program or transfer of a file of size $\ell$ in presence of failures (epochs of the process) which necessitate restart.

math.PR↗

arXiv2016

Exit identities for Lévy processes observed at Poisson arrival times

For a spectrally one-sided Lévy process, we extend various two-sided exit identities to the situation when the process is only observed at arrival epochs of an independent Poisson process. In addition, we consider exit problems of this type for processes reflected either from above or from below. The resulting Laplace transforms of the main quantities of interest are in terms of scale functions and turn out to be simple analogues of the classical formulas.

math.PR↗

arXiv2015

Splitting and time reversal for Markov additive processes

We consider a Markov additive process with a finite phase space and study its path decompositions at the times of extrema, first passage and last exit. For these three families of times we establish splitting conditional on the phase, and provide various relations between the laws of post- and pre-splitting processes using time reversal. These results offer valuable insight into behavior of the process, and while being structurally similar to the Lévy process case, they demonstrate various new features. As an application we formulate the Wiener-Hopf factorization, where time is counted in each phase separately and killing of the process is phase dependent. Assuming no positive jumps, we find concise formulas for these factors, and also characterize the time of last exit from the negative half-line using three different approaches, which further demonstrates applicability of path decomposition theory.

math.PR↗

arXiv2015

Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations

We consider exit problems for general Lévy processes, where the first passage over a threshold is detected either immediately or at an epoch of an independent homogeneous Poisson process. It is shown that the two corresponding one-sided problems are related through a surprisingly simple identity. Moreover, we identify a simple link between two-sided exit problems with one continuous and one Poisson exit. Finally, Poisson exit of a reflected process is connected to the continuous exit of a process reflected at Poisson epochs, and a link between some Parisian type exit problems is established. With the appropriate perspective, the proofs of all these relations turn out to be quite elementary. For spectrally one-sided Lévy processes this approach enables alternative proofs for a number of previously established identities, providing additional insight.

math.PR↗

arXiv2015

Sparre-Andersen identity and the last passage time

It is shown that the celebrated result of Sparre Andersen for random walks and Lévy processes has intriguing consequences when the last time of the process in $(-\infty,0]$, say $σ$, is added to the picture. In the case of no positive jumps this leads to six random times, all of which have the same distribution - the uniform distribution on $[0,σ]$. Surprisingly, this result does not appear in the literature, even though it is based on some classical observations concerning exchangeable increments.

math.PR↗

arXiv2015

Transient analysis of a stationary Lévy-driven queue

In this paper we study a queue with Lévy input, without imposing any a priori assumption on the jumps being one-sided. The focus is on computing the transforms of all sorts of quantities related to the transient workload, assuming the workload is in stationarity at time 0. The results are simple expressions that are in terms of the bivariate Laplace exponents of ladder processes. In particular, we derive the transform of the minimum workload attained over an exponentially distributed interval.

math.PR↗

arXiv2015

Lévy-driven polling systems and continuous-state branching processes

In this paper we consider a ring of $N\ge 1$ queues served by a single server in a cyclic order. After having served a queue (according to a service discipline that may vary from queue to queue), there is a switch-over period and then the server serves the next queue and so forth. This model is known in the literature as a \textit{polling model}. Each of the queues is fed by a non-decreasing Lévy process, which can be different during each of the consecutive periods within the server's cycle. The $N$-dimensional Lévy processes obtained in this fashion are described by their (joint) Laplace exponent, thus allowing for non-independent input streams. For such a system we derive the steady-state distribution of the joint workload at embedded epochs, i.e. polling and switching instants. Using the Kella-Whitt martingale, we also derive the steady-state distribution at an arbitrary epoch. Our analysis heavily relies on establishing a link between fluid (Lévy input) polling systems and multi-type Jiřina processes (continuous-state discrete-time branching processes). This is done by properly defining the notion of the \textit{branching property} for a discipline, which can be traced back to Fuhrmann and Resing. This definition is broad enough to contain the most important service disciplines, like exhaustive and gated.

math.PR↗

arXiv2015

A bivariate risk model with mutual deficit coverage

We consider a bivariate Cramer-Lundberg-type risk reserve process with the special feature that each insurance company agrees to cover the deficit of the other. It is assumed that the capital transfers between the companies are instantaneous and incur a certain proportional cost, and that ruin occurs when neither company can cover the deficit of the other. We study the survival probability as a function of initial capitals and express its bivariate transform through two univariate boundary transforms, where one of the initial capitals is fixed at 0. We identify these boundary transforms in the case when claims arriving at each company form two independent processes. The expressions are in terms of Wiener-Hopf factors associated to two auxiliary compound Poisson processes. The case of non-mutual (reinsurance) agreement is also considered.

math.PR↗

arXiv2014

A Lévy process on the real line seen from its supremum and max-stable processes

We consider a process $Z$ on the real line composed from a Lévy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of $Z$ seen from its supremum, the supremum $\overline Z$ and the time $T$ at which the supremum occurs. In fact, it is closely related to the laws of the original and the tilted Lévy processes conditioned to stay negative and positive. The result is used to derive a new representation of stationary particle systems driven by Lévy processes. In particular, this implies that a max-stable process arising from Lévy processes admits a mixed moving maxima representation with spectral functions given by the conditioned Lévy processes.

math.PR↗

arXiv2014

Power identities for Lévy risk models under taxation and capital injections

In this paper we study a spectrally negative Lévy process which is refracted at its running maximum and at the same time reflected from below at a certain level. Such a process can for instance be used to model an insurance surplus process subject to tax payments according to a loss-carry-forward scheme together with the flow of minimal capital injections required to keep the surplus process non-negative. We characterize the first passage time over an arbitrary level and the cumulative amount of injected capital up to this time by their joint Laplace transform, and show that it satisfies a simple power relation to the case without refraction. It turns out that this identity can also be extended to a certain type of refraction from below. The net present value of tax collected before the cumulative injected capital exceeds a certain amount is determined, and a numerical illustration is provided.

q-fin.PR↗

arXiv2013

A risk model with an observer in a Markov environment

We consider a spectrally-negative Markov additive process as a model of a risk process in random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process negative, where the observation rate may depend on the state of the environment. Using an approximation argument and spectral theory we establish an explicit formula for the resulting survival probabilities in this general setting. We also discuss an efficient evaluation of the involved quantities and provide a numerical illustration.

math.PR↗

arXiv2013

Potential measures of one-sided Markov additive processes with reflecting and terminating barriers

Consider a one-sided Markov additive process with an upper and a lower barrier, where each can be either reflecting or terminating. For both defective and non-defective processes and all possible scenarios we identify the corresponding potential measures, which generalizes a number of results for one-sided Levy processes. The resulting rather neat formulas have various applications, and in particular they lead to quasi-stationary distributions of the corresponding processes.

math.PR↗

arXiv2013

Exact boundaries in sequential testing for phase-type distributions

We consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. In this setting, we derive exact decision boundaries for given Type I and Type II errors by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated to a certain one-sided Markov additive process. By suitable transformations the results also apply to other types of distributions including some distributions with regularly varying tail.

math.PR↗

arXiv2013

Two coupled Levy queues with independent input

We consider a pair of coupled queues driven by independent spectrally-positive Levy processes. With respect to the bi-variate workload process this framework includes both the coupled processor model and the two-server fluid network with independent Levy inputs. We identify the joint transform of the stationary workload distribution in terms of Wiener-Hopf factors corresponding to two auxiliary Levy processes with explicit Laplace exponents. We reinterpret and extend the ideas of Cohen and Boxma (1983) to provide a general and uniform result with a neat transform expression.

math.PR↗

arXiv2011

Occupation densities in solving exit problems for Markov additive processes and their reflections

This paper solves exit problems for spectrally negative Markov additive processes and their reflections. A so-called scale matrix, which is a generalization of the scale function of a spectrally negative \levy process, plays a central role in the study of exit problems. Existence of the scale matrix was shown in Thm. 3 of Kyprianou and Palmowski (2008). We provide a probabilistic construction of the scale matrix, and identify the transform. In addition, we generalize to the MAP setting the relation between the scale function and the excursion (height) measure. The main technique is based on the occupation density formula and even in the context of fluctuations of spectrally negative Lévy processes this idea seems to be new. Our representation of the scale matrix $W(x)=e^{-Λx}\eL(x)$ in terms of nice probabilistic objects opens up possibilities for further investigation of its properties.

math.PR↗