Jevgenijs Ivanovs

Jevgenijs Ivanovs
Aarhus University | AU · Department of Mathematics

PhD

About

74
Publications
3,620
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780
Citations
Citations since 2016
35 Research Items
584 Citations
2016201720182019202020212022020406080100
2016201720182019202020212022020406080100
2016201720182019202020212022020406080100
2016201720182019202020212022020406080100

Publications

Publications (74)
Article
Full-text available
This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian...
Article
We consider two-dimensional Lévy processes reflected to stay in the positive quadrant. Our focus is on the non-standard regime when the mean of the free process is negative but the reflection vectors point away from the origin, so that the reflected process escapes to infinity along one of the axes. Under rather general conditions, it is shown that...
Article
Full-text available
We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations of a Lévy process. The first procedure relies on reordering of independently sampled normal increments and thus avoids tuning parameters. The functionality of this method is a consequence of the small time predom...
Preprint
Full-text available
We introduce a new method analyzing the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive...
Article
Full-text available
We provide a simple algorithm for construction of Brownian paths approximating those of a Lévy process on a finite time interval. It requires knowledge of the Lévy process trajectory on a chosen regular grid and the law of its endpoint, or the ability to simulate from that. This algorithm is based on reordering of Brownian increments, and it can be...
Article
There is an abundance of useful fluctuation identities for one-sided Lévy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced by...
Article
Full-text available
We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical...
Preprint
Full-text available
This paper provides a multivariate extension of Bertoin's pathwise construction of a L\'evy process conditioned to stay positive/negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian m...
Article
This paper considers discretization of the Lévy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity assumptions on the given Lévy process. Additionally, the scaling limit of a positive self-similar Markov process at...
Preprint
Full-text available
We provide a simple algorithm for construction of Brownian paths approximating those of a L\'evy process on a finite time interval. It requires knowledge of the L\'evy process trajectory on a chosen regular grid and the law of its endpoint, or the ability to simulate from that. This algorithm is based on reordering of Brownian increments, and it ca...
Article
Extreme value statistics provides accurate estimates for the small occurrence probabilities of rare events. While theory and statistical tools for univariate extremes are well developed, methods for high-dimensional and complex data sets are still scarce. Appropriate notions of sparsity and connections to other fields such as machine learning, grap...
Preprint
There is an abundance of useful fluctuation identities for one-sided L\'evy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced...
Preprint
Full-text available
Tukey's halfspace depth can be seen as a stochastic program and as such it is not guarded against optimizer's curse, so that a limited training sample may easily result in a poor out-of-sample performance. We propose a generalized halfspace depth concept relying on the recent advances in distributionally robust optimization, where every halfspace i...
Preprint
We provide a novel expression of the scale function for a L\'evy processes with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and analyzed. Our numerical examples suggest that this algorithm al...
Preprint
We consider two-dimensional L\'evy processes reflected to stay in the positive quadrant. Our focus is on the non-standard regime when the mean of the free process is negative but the reflection vectors point away from the origin, so that the reflected process escapes to infinity along one of the axes. Under rather general conditions, it is shown th...
Preprint
Full-text available
This paper considers discretization of the L\'evy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity assumptions on the given L\'evy process. Additionally, the scaling limit of a positive self-similar Markov process...
Article
We consider two bivariate models with two-way interactions in context of risk and queueing theory. The two entities interact with each other by providing assistance but otherwise evolve independently. We focus on certain random quantities underlying the joint survival probability and the joint stationary workload, and show that these admit stochast...
Preprint
Extreme value statistics provides accurate estimates for the small occurrence probabilities of rare events. While theory and statistical tools for univariate extremes are well-developed, methods for high-dimensional and complex data sets are still scarce. Appropriate notions of sparsity and connections to other fields such as machine learning, grap...
Preprint
Full-text available
We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations of a Lévy process. The first procedure relies on reordering of independently sampled normal increments and thus avoids tuning parameters. The main reasons underlying this method are small time predominance of the...
Preprint
In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a L\'evy process. More specifically, we investigate the asymptotic theory for the conditional mean and conditional median estimators of the supremum/infimum of a linear Brownian motion and a stable L\'evy proce...
Preprint
We consider two bivariate models with two-way interactions in context of risk and queueing theory. The two entities interact with each other by providing assistance but otherwise evolve independently. We focus on certain random quantities underlying the joint survival probability and the joint stationary workload, and show that these admit stochast...
Preprint
For a L\'evy process $X$ on a finite time interval consider the probability that it exceeds some fixed threshold $x>0$ while staying below $x$ at the points of a regular grid. We establish exact asymptotic behavior of this probability as the number of grid points tends to infinity. We assume that $X$ has a zooming-in limit, which necessarily is $1/...
Article
This note provides a factorization of a Lévy pocess over a phase-type horizon τ given the phase at the supremum, thereby extending the Wiener–Hopf factorization for τ exponential. One of the factors is defined using time reversal of the phase process. It is shown that there are a variety of time-reversed representations, all yielding the same facto...
Article
Full-text available
An obvious way to simulate a L\'evy process $X$ is to sample its increments over time $1/n$, thus constructing an approximating random walk $X^{(n)}$. This paper considers the error of such approximation after the two-sided reflection map is applied, with focus on the value of the resultant process $Y$ and regulators $L,U$ at the lower and upper ba...
Article
In the context of collective risk theory, we give a sample path identity relating capital injections in the original model and dividend payments in the time-reversed counterpart. We exploit this duality to provide an alternative view on some of the known results on the expected discounted capital injections and dividend payments for risk models dri...
Article
Full-text available
This note provides a factorization of a L\'evy pocess over a phase-type horizon $\tau$ given the phase at the supremum, thereby extending the Wiener-Hopf factorization for $\tau$ exponential. One of the factors is defined using time reversal of the phase process. It is shown that there are a variety of time-reversed representations, all yielding th...
Article
Full-text available
We study the joint distribution of tax payments according to a loss-carry forward scheme and capital injections in a Lévy risk model, and provide a transparent expression for the corresponding transform in terms of the scale function. This allows to identify the net present value of capital injections in such a model, complementing the one for tax...
Article
Full-text available
Let $L_t$ be the longest gap before time $t$ in an inhomogeneous Poisson process with rate function $\lambda_t$ proportional to $t^{\alpha-1}$ for some $\alpha\in(0,1)$. It is shown that $\lambda_tL_t-b_t$ 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...
Article
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-side...
Article
Let $M$ and $\tau$ be the supremum and its time of a L\'evy process $X$ on some finite time interval. It is shown that zooming in on $X$ at its supremum, that is, considering $(a_\eta(X_{\tau+t/\eta}-M))_{t\in\mathbb R}$ as $\eta,a_\eta\rightarrow\infty$, results in $(\xi_t)_{t\in\mathbb R}$ constructed from two independent processes corresponding...
Article
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 modelled by a parametric family of spectral distributions. In this work we provide...
Article
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 (-∞, 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]. Surprisin...
Article
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 the supremum (image found), its time T, and the process Z(T + ·) –(image found). This expression is in terms of the laws of the original and the tilted Lévy processes cond...
Article
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 resu...
Article
Full-text available
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...
Article
In this paper we study a queue with L\'evy 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 Lapl...
Article
Full-text available
It is shown that the celebrated result of Sparre Andersen for random walks and L\'evy processes has intriguing consequences when the last time of the process in $(-\infty,0]$ 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.
Article
We consider a bivariate 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...
Article
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 inj...
Article
Full-text available
We consider a process $Z$ on the real line composed from a L\'evy 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 th...
Article
Full-text available
For a spectrally one-sided \levy 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 from above or from below. The resulting transforms of the main quantities of interest...
Article
Full-text available
Taxed risk processes, i.e. processes which change their drift when reaching new maxima, represent a certain type of generalizations of Lévy and of Markov additive processes (MAP), since the times at which their Markovian mechanism changes are allowed to depend on the current position. In this paper we study generalizations of the tax identity of Al...
Article
Full-text available
In this note we provide a simple alternative probabilistic derivation of an explicit formula of I. K. M. Kwan and H. Yang [“Ruin probability in a threshold insurance risk model”, Belg. Actuar. Bull. 7, 41–49 (2007), http://www.belgianactuarialbulletin.be/browse.php?issue=77-8] for the probability of ruin in a risk model with a certain dependence be...
Article
We consider a pair of coupled queues driven by independent spectrally-positive Lévy 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 Lévy inputs. We identify the joint transform of the stationary workload distribution in terms of Wie...
Article
Full-text available
In this paper we study a spectrally negative L\'evy 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 i...
Article
Full-text available
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 approximat...
Article
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 nondefective processes, and all possible scenarios, we identify the corresponding potential measures, which help to generalize a number of results for one-sided Lévy processes. The resulting rath...
Article
In this note, we identify a simple setup from which one may easily infer various decomposition results for queues with interruptions as well as càdlàg processes with certain secondary jump inputs. Special cases are processes with stationary or stationary and independent increments. In the Lévy process case, the decomposition holds not only in the l...
Article
Full-text available
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...
Article
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 Wie...
Article
It is often natural to consider defective or killed stochastic processes. Various observations continue to hold true for this wider class of processes yielding more general results in a transparent way without additional effort. We illustrate this point with an example from risk theory by showing that the ruin probability for a defective risk proce...
Article
Abstract Human pose recognition is an important problem in the computer vision domain. Pose recognition can be used as a basis for surveillance, human computer interaction and motion analysis systems, provided recognition is done reliably and efficiently. A 2D approach with explicit shape model is used in this work. More specifically, we consider a...
Article
This article considers a Markov-modulated Brownian motion with a two-sided reflection. For this doubly-reflected process we compute the Laplace transform of the stationary distribution, as well as the average loss rates at both barriers. Our approach relies on spectral properties of the matrix polynomial associated with the generator of the free (t...
Article
Full-text available
In this paper we consider a ring of N ≥ 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 polling model. Each of the q...
Article
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 (2...
Article
In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions....
Article
We consider a Markov-modulated Brownian motion reflected to stay in a strip [0, B ]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explain its simplicity. Moreover, this argument allows for generalizations including the distribu...
Article
Full-text available
In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions....
Article
We analyze the number of zeros of det(F(α)), where F(α) is the matrix exponent of a Markov Additive Process (MAP) with one-sided jumps. The focus is on the number of zeros in the right half of the complex plane, where F(α) is analytic. In addition, we also consider the case of a MAP killed at an independent exponential time. The corresponding zeros...
Article
Full-text available
In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions....
Article
Full-text available
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}. E...
Article
Full-text available
We present a new approach to fluctuation identities for reflected L\'{e}vy processes with one-sided jumps. This approach is based on a number of easy to understand observations and does not involve excursion theory or It\^{o} calculus. It also leads to more general results. Comment: 6 pages
Article
Full-text available
We consider a Markov-modulated Brownian motion reflected to stay in a strip [0,B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explaining its simplicity. Moreover, this argument allows for generalizations including the distrib...
Article
We study the first passage process of a spectrally negative Markov additive process (MAP). The focus is on the background Markov chain at the times of the first passage. This process is a Markov chain itself with a transition rate matrix Λ. Assuming time reversibility, we show that all the eigenvalues of Λ are real, with algebraic and geometric mul...
Article
Full-text available
In this paper we consider a ring of N ≥ 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 polling model. Each of the q...
Article
We study the record process of a spectrally-negative Markov additive process (MAP). Assuming time-reversibility, a number of key quantities can be given explicitly. It is shown how these key quantities can be used when analyzing the distribution of the all-time maximum attained by MAPs with negative drift, or, equivalently, the stationary workload...
Article
Full-text available
We analyze the number of zeros of det(F(alpha)), where F(alpha) is the matrix cumulant generating function of a Markov Additive Process (MAP) with one-sided jumps. The focus is on the number of zeros in the right half of the comple x plane, where det(F(alpha)) is well-defined. Moreover, we analyze the case of a killed MAP with state-dependent killi...
Conference Paper
Full-text available
We consider the problem of scoring Bayesian Network Classifiers (BNCs) on the basis of the conditional loglikelihood (CLL). Currently, optimization is usually performed in BN parameter space, but for perfect graphs (such as Naive Bayes, TANs and FANs) a mapping to an equivalent Logistic Regression (LR) model is possible, and optimization can be per...
Article
The complexity of quantum query algorithms computing Boolean functions is strongly related to the degree of the algebraic polynomial representing this Boolean function. There are two related difficult open problems. First, Boolean functions are sought for which the complexity of exact quantum query algorithms is essentially less than the complexity...
Conference Paper
Full-text available
The complexity of quantum query algorithms computing Boolean functions is strongly related to the degree of the algebraic polynomial representing this Boolean function. There are two related difficult open problems. First, Boolean functions are sought for which the complexity of exact quantum query algorithms is essentially less than the complexity...
Article
Full-text available
Lèvy-processen vormen een klassiek model uit de toegepaste kansrekening, en worden gekenmerkt door stationaire en onafhankelijke incrementen. Ze kennen vele toepassingen, bijvoorbeeld bij het modelleren van biologische processen, voorraadsystemen, bij risicomanagement van verzekeringen en binnen de financiële wiskunde in het algemeen. Veel processe...
Article
Full-text available
In this paper we consider the two-sided reflection of a Markov modulated Brownian motion by analyzing the spectral properties of the matrix polynomial associated with the generator of the free process. We show how to compute for the general case the Laplace transform of the stationary distribution and the average loss rates at both barriers for the...

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