O seminarium

Terminy i tematyka spotkań

czwartek, 30-03-2017 - 12:15, 602
A convergence theorem for stationary heavy tailed sequences
Azra Tafro (University of Zagreb)
Point processes theory is a useful tool for the extremal analysis of stochastic processes. It is well known, for instance, that for an iid sequence of random variables, regular variation of the marginal distribution is equivalent to the convergence of the point process generated by the sequence towards a suitable Poisson point process. That statement then yields many asymptotic distributional properties about the original sequence. There are many extensions of this result to dependent stationary sequences and the point processes generated by them. We will give such a result for a class of weakly dependent regularly varying processes in the multivariate setting. As an application of the result, we will show the invariance principle for the so-called maximal process. This is joint work with Bojan Basrak.
czwartek, 16-03-2017 - 12:15, 602
A drawdown-based regime-switching Lévy insurance model
Shu Li (University of Illinois)
In this talk, we will talk about a drawdown-based regime-switching (DBRS) Lévy insurance model in which the underlying drawdown process is used to model an insurer’s level of financial distress over time, and to trigger regime-switching transitions. By some analytical arguments, we derive explicit formulas for a generalized two-sided exit problem. We specifically state the positive security loading conditions. The regime-dependent occupation time until ruin is later studied. As a special case of the general DBRS model, a regime-switching premium model is given further consideration. Connections with other existing risk models are established.
czwartek, 09-03-2017 - 12:15, 602
On the optimal dividend problem with transaction cost and parisian ruin time for a refracted Lévy process
Adam Kaszubowski (Uniwersytet Wrocławski)
We consider optimal dividend problem for a company, whose risk process is modeled by refracted Levy process. At the beginning I will recall the idea of dividends and historical stochastic approach to this problem. Afterwards I will formally define model, problem and dividend strategy, which will be a candidate for being the optimal one. In our setup we choose ruin time to be parisian, which means that ruin occurs when risk process is for the first time below zero longer than some constant $ r > 0$. Additionally we assume that after every dividend, company must pay fixed transaction cost, this causes that classical barrier strategy cannot be optimal anymore. All results will be written in terms of the scale functions and numerical examples of this results will be shown.
czwartek, 23-02-2017 - 12:15, 602
Drawdown insurance contracts
Joanna Tumilewicz (Uniwersytet Wrocławski)
We consider some drawdown insurance contracts with constant permium rate, general reward and penalty functions. The drawdown proces we define as difference between historical maximum and current asset value. We focus on two problem: calculating the fair premium for basic contracts and finding the optimal stopping rule for polices with additional cancellable feature. We use the fluctuation theory of Lévy processes and theory of optimal stopping.
czwartek, 02-02-2017 - 12:15, 602
Multi-refracted Lévy risk processes
Irmina Czarna (Uniwersytet Wrocławski)

We consider multi-refracted Lévy risk process whose dynamics change by subtracting off a fixed linear drifts whenever the process is above certain levels. Formally, we define a multi-refracted Lévy risk process as a unique strong solution of the SDE (for $k \geq 1$): $$\mathrm{d}U_k(t) = \mathrm{d}X(t) - \left(\delta_1 \mathbf{1}_{\{U_k(t) > b_1\}}+\delta_2 \mathbf{1}_{\{U_k(t) > b_2\}}+...+\delta_k \mathbf{1}_{\{U_k(t) > b_k\}} \right)\mathrm{d}t , \quad t \geq 0 ,$$ where $X$ is a spectrally negative Lévy process, $\delta_1,\ldots,\delta_k$ and $b_1 < b_2 < \ldots < b_n$ are model parameters.

Moreover, we present the formulas for one and two sided exit problems written in terms of the new $q$- scale functions associate with the process $U_k$. We also present new properties of the obtained scale functions. Finally, we extend the theory of multi-refracted processes to processes with general premium rate function $\phi$.

czwartek, 19-01-2017 - 12:15, 602
On classical queueing networks
Ryszard Szekli (Uniwersytet Wrocławski)

I will recall classical mathematical problems related to queueing networks, some classical results related to ergodicity of some related Markov processes, and will discuss questions on the speed of convergence to stationarity and ordering of networks.

The talk will be based on some papers with Hans Daduna or Pawel Lorek, including recent ones but also some older ones, for example:

Correlation formulas for Markovian network processes in a random environment. Adv. in Appl. Probab. 48 (2016)

Computable bounds on the spectral gap for unreliable Jackson networks. Adv. in Appl. Probab. 47 (2015)

Stochastic comparison of queueing networks. Queueing networks, 345–395, Internat. Ser. Oper. Res. Management Sci., 154, Springer, New York, 2011

czwartek, 12-01-2017 - 12:15, 602
Siegmund duality, antiduality and Fastest Strong Stationary Times.
Paweł Lorek
We recall construction of Siegmund dual chain for Mobius monotone Markov chains. In first part of the talk we present 3 types of application of the relation between ergodic chain and its Siegmund dual: A1) solving ruin-like problems; A2) finding stationary distribution; A3) estimating stationary distribution via "stable simulation scheme". In second part of the talk we show how to find Fastest Strong Stationary Time (FSST) of a chain using Siegmund dual chain. The construction of a chain with prescribed FSST will be presented. Exploiting existing results on limiting distribution of "time till all coupons are collected" in several generalizations of coupon collector problem, we present chain exhibiting so-called separation cutoff. The talk will be accompanied by numerous examples.
czwartek, 08-12-2016 - 12:15, 602
Drift change detection in stochastic mortality rate models.
Michał Krawiec

We will consider the theory of optimal change point detection of stochastic processes applied to the mortality rate models. At the beginning there will be some motivations and general description of the topic. Then we will proceed to the first detection model based on Brownian motion, for which at some random time drift changes. The first part of the talk will focus on this model and its applications in analysing Polish life tables. In the second part of the talk we will consider other detection models for Poisson and compound Poisson processes. We will also see some theoretical results for the process consisting of both continuous and jump part.

The talk is based on our joint work with Prof. Z. Palmowski.
czwartek, 01-12-2016 - 12:15, 602
Extremal Markovian sequences of the Kendall type.
Barbara Jasiulis-Gołdyn
W załączonym pliku.
czwartek, 24-11-2016 - 12:15, 602
Random matrix approximation for some non-commutative stochastic processes.
José Luis Pérez Garmendia
A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measurevalued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu. Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used. Also the case of a Non-commutative Fractional Poisson Process will be discussed, in terms of an approximation based on the fractional Wishart process.
czwartek, 10-11-2016 - 12:15, 602
Branching process in random environment an related models.
Piotr Dyszewski

We will discuss one of many population growth models, namely the branching process in random environment. We will show how asymptotic properties of perpetuities affect those of our model. If time allows, we will show some applications to random walks in random environment.

The talk is based mostly on the work of Kesten, Kozlov & Spitzer [1975, Compositio Mathematica].
czwartek, 27-10-2016 - 12:15, 602
Limit theory for geometric statistics of clustering point processes
Bartłomiej Błaszczyszyn

(joint work with D. Yogeshwaran and J. E. Yukich)

Let $P$ be a simple, stationary, clustering point process on the $d$-dimensional Euclidean space, in the sense that its correlation functions factorize up to an additive error decaying exponentially fast with the separation distance. Let $P_n$ be its restriction to the windows of volume $n$. We consider statistics of $P_n$ admitting the representation as sums of spatially dependent terms $H_n=\sum_{x\in P_n} \xi(x,P_n)$, where $\xi(x,P_n)$ is a real valued (score) function, representing the interaction of $x$ with $P_n$. When the score function depends locally on $P_n$ in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and central limit theorems for $H_n$ as the volume n of the window goes to infinity.

This gives the limit theory for non-linear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model, and total edge length of the $k$-nearest neighbor graph) of determinantal point processes with fast decreasing kernels, including the $\alpha$-Ginibre ensembles. It also gives the limit theory for geometric $U$-statistics of permanental point processes as well as the zero set of Gaussian entire functions. This extends the existing literature treating the limit theory of sums of stabilizing scores of Poisson and binomial input. In the setting of clustering point processes, it also extends the results of Soshnikov (2002) as well as work of Nazarov and Sodin (2012).

The proof of the central limit theorem relies on a factorial moment expansion originating in Blaszczyszyn (1995) to show clustering of mixed moments of the score function. Clustering extends the cumulant method to the setting of purely atomic random measures, yielding the asymptotic normality of $H_n$.

czwartek, 20-10-2016 - 12:15, 602
Multivariate models connected with random sums and maxima.
Marek Arendarczyk

In this talk I will consider a stochastic model for $(X,Y,N)$, where $X$ and $Y$ respectively, are the sum and the maximum of $N$ dependent, heavy-tailed Pareto components. Models with this or similar structure are desirable in many applications, ranging from hydro-climatology to finance and insurance. Theoretical results, as well as real data examples, illustrating the usefulness of the model will be presented.

The talk is based on joint work with Tomasz J. Kozubowski and Anna K. Panorska (University of Nevada, Reno).
czwartek, 13-10-2016 - 12:15, 602
Metoda couplingu w badaniu szybkości zbieżności.
Piotr Markowski
Badając prędkość zbieżność rozkładu procesu Markowa do rozkładu stacjonarnego możemy korzystać z metody couplingowej. Okazuje się, że znalezienie markowskiego couplingu (tzn. pary funkcji losowych (X,Y) będącej procesem Markowa) nie zawsze prowadzi do optymalnego oszacowania szybkości zbieżności. Prelegent przypomni kilka znanych i zaprezentuje kilka nowych konstrukcji couplingowych.
czwartek, 06-10-2016 - 12:15, 602
O stałych Pickandsa.
Krzysztof Dębicki

Prelegent opowie o różnych reprezentacjach i kontekstach, gdzie znaleziono te stałe.

Wystąpienie oparte będzie o wspólne prace z Enkelejdem Hashorvą i Sebastianem Engelke.
czwartek, 12-05-2016 - 12:15, 602
Prawdopodobieństwo ruiny dla skorelowanych ruchów Browna.
Krzysztof Dębicki

Niech $\boldsymbol{B}(t)=(B_1(t),\ldots,B_d(t))^{'},t\ge0$ będzie standardowym $d$-wymiarowym ruchem Browna. W prezentacji zanalizujemy dokładną asymptotykę $$ \mathbb{P}\left(\exists\ {t\ge 0}: \boldsymbol{AX}(t)- \boldsymbol{\mu}t> \boldsymbol{\alpha} u \right), $$ gdy $u\to\infty$, gdzie $A$ jest macierzą niesingularną oraz wektory $\boldsymbol{\alpha}, \boldsymbol{\mu}$ spełniają $\boldsymbol{\alpha}>\boldsymbol{0},$ $\boldsymbol{\mu}$, $\max_{1\le i \le d} \mu_i>0$. Dodatkowo zbadamy własności wielowymiarowych odpowiedników stałych Pickands'a, które pojawiają się w uzyskanych asymptotykach.

Wykład oparty jest o wyniki wypracowane wspólnie z E. Hashorwą (Univ. of Lausanne), L. Ji (Univ. of Lausanne) oraz T. Rolskim.
czwartek, 28-04-2016 - 12:15, 602
Problemy wyjścia dla spadków i wzrostów procesów Lévy'ego w wycenie kontraktów ubezpieczeniowych.
Joanna Tumilewicz
/streszczenie w załączniku/
czwartek, 28-04-2016 - 12:15, 602
Fluktuacje spektralnie ujemnego procesu Lévy'ego zabijanego z intensywnością zależną od stanu tego procesu.
Zbigniew Palmowski

W referacie rozwiażemy tzw. problemy wyjścia dla (odbitego w supremum lub infimum) spektralnie ujemnego procesu Lévy'ego $X$ wykładniczo zabijanego z intensywnością będącą pewną funkcją $\omega$ od obecnego stanu tego procesu. Wszystkie tożsamości będą wyrażone poprzez uogólnienia klasycznych funkcji skalujących.

Referat będzie oparty o pracę: B. Li i Z. Palmowski, Fluctuations of Omega-killed spectrally negative Lévy processes, 2016. Złożony do publikacji. http://arxiv.org/abs/1603.07967.
czwartek, 07-04-2016 - 12:15, 602
Paryskie problemy wyjścia dla rozszczepionego procesu Lévy'ego
Irmina Czarna

Podczas referatu zdefiniujemy tzw. rozszczepiony proces Lévy'ego, czyli proces, który zmienia swoje zachowanie w zależności czy jest powyżej czy poniżej z góry określonej bariery. Dla tego procesu wyznaczymy prawdopodobieństwo paryskiej ruiny oraz rozwiążemy tzw. jednostronne i dwustronne problemy wyjścia rozważane do momentu paryskiej ruiny. Przypomnijmy, że paryskie opóźnienie, oznacza zadany z góry, deterministyczny czas $r>0$, przez który badany proces Lévy'ego musi znajdować się w określonym położeniu: na przykład powyżej/poniżej pewnego poziomu (tak zwanej bariery). Natomiast paryska ruina to pierwszy moment kiedy rozważany proces ryzyka pozostaje poniżej zera dłużej niż ustalony horyzont czasowy $r>0$. Prezentowane rezultaty uogólniają te otrzymane w pracy [1]. Ponadto podczas referatu przeanalizujemy otrzymane formuły dla szczególnych przykładów procesów Lévy'ego.

[1] R. Loeffen, I. Czarna, Z. Palmowski, Parisian ruin probability for spectrally negative Lévy processes, Bernoulli 2013, Vol. 19, No. 2, 599-609.
czwartek, 17-03-2016 - 12:15, 602
Wykrywanie punktu zmiany w dwufazowym modelu typu Lévy'ego
Michał Krawiec
Rozważać będziemy proces $X_t$ ulegający zmianie w losowej, nieobserwowalnej chwili $\theta$ o zadanym z góry rozkładzie. To znaczy proces $X_t$ definiujemy w następujący sposób: $X_t = X_t^{(1)}$ dla $t<\theta$ oraz $X_t = X_t^{(2)}$ dla $t>\theta$, gdzie oba procesy $X_t^{(i)}$ ($i=1,2$) są zadanymi procesami Lévy'ego związanymi wykładniczą zamianą miary. Problem wykrywania punktu zmiany procesu rozpatrzymy pod kątem znalezienia optymalnego czasu zatrzymania $\tau$ minimalizującego kryterium postaci: $\inf\{P(\tau<\theta) + cE[(\tau-\theta)^+]\}$. Kluczową rolę dla rozwiązania problemu będzie miał proces $\pi_t$ oparty o prawdopodobieństwo a posteriori względem naturalnej filtracji generowanej przez $X_t$. Znajdziemy generator Markowskiego procesu $\pi_t$ oraz przy jego użyciu sprowadzimy problem optymalnej detekcji do rozwiązania pewnego problemu wariacyjnego równoważnego identyfikacji największego submartyngału spełniającego pewne warunki brzegowe. Powyższa analiza zostanie poparta przykładami związanymi z ruchem Browna i procesem Poissona.
czwartek, 10-03-2016 - 12:15, 602
Extremes of transient Gaussian fluid queues.
Peng Liu
Let $\{X(t),t\geq 0\}$ be a centered Gaussian process with stationary increments, a.s. continuous sample paths and variance function $\sigma^2(t)$. Given $c>0$, we consider the fluid queue \[ Q_x(t):=\max\left(x+X(t)-ct, \sup_{0\le s\le t}\left(X(t)-X(s)-c(t-s)\right)\right),\ \ \ t >0, \] where $X(t)-X(s)$ denotes the total inflow to the system in time interval $(s,t]t$, $c$ is the service rate and $x=Q_x(0)$ . The talk is focused on the asymptotic behaviour of $P(Q_x(T_u)>u) $ and $P(\sup_{t\in[0,T_u]} Q_x(t)>u) $ as $u\to \infty$.
czwartek, 25-02-2016 - 12:15, 602
Siegmund duality for Markov chains on partially ordered state spaces: Generalized Gambler's Ruin Problem
Paweł Lorek

For Markov chains on finite partially ordered state space we show that Siegmund dual exists if and only if the chain is Möbius monotone, in which case we give formula for its transitions. Exploiting the relation between ergodic Markov chain and its Siegmund dual we give a procedure for solving ruin-like problems. As main application we give solution for ruin probability in some Generalized Gambler's Ruin Problem. The generalization is two-folded: i) winning/losing probabilities depend on the  current capital; ii) it is multidimensional, i.e., involves many players.  We also show how to construct a Strong Stationary Dual chain (with link being truncated stationary distribution) by performing appropriate Doob transform of the Siegmund dual of time-reversed chain.

czwartek, 21-01-2016 - 12:15, 602
Wkład prof. C.Ryll-Nardzewskego do probabilistyki, w szczególności do procesów punktowych
Tomasz Rolski
czwartek, 14-01-2016 - 12:15, 602
Estimates of Dirichlet heat kernel for symmetric Markov processes.
Kyung-youn Kim

We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal L$\acute{e}$vy processes with weak scaling conditions. We first establish sharp two-sided heat kernel estimates for these processes in $C^{1,\rho}$ open sets, $\rho \in (\overline{\alpha}/2, 1]$ where $\overline{\alpha}$ is the upper scaling parameter in the weak scaling conditions. As a corollary of our main result, we obtain a sharp two-sided Green function estimates and a scale invariant boundary Harnack inequality with explicit decay rates in $C^{1,\rho}$ open sets.

czwartek, 17-12-2015 - 12:15, 602
Smoothing transform and thin tails.
Piotr Dyszewski

Consider a (canonical) solution to the stochastic fixed point equation $X =^d \sum_{k=1}^NT_k X_k +C$, where $X, X_1, X_2 ...$ are iid random variables independent of the random vector $(C, T_1, T_2 ...)$. We will provide conditions on $(C, T_1 , T_2 ...)$ such that this solution exhibits right and/or left Poisson tails and the abscissa of convergence of its moment generating function can be determined. As a particular application, the right tail behavior of the Quicksort distribution is found. The talk is based on a joint work with Gerold Alsmeyer (University of Münster).

czwartek, 29-10-2015 - 12:15, 602
Limit theory for geometric statistics of clustering point processes
dr hab. Bartłomiej Błaszczyszyn

Let $\P$ be a clustering point process on $\R^d$ and let $\P_n:= \P \cap W_n$ be its restriction to windows $W_n \subset \R^d$. We consider the statistic $H_n^\xi:= \sum_{x \in \P_n} \xi(x, \P_n)$ where $\xi(x, \P_n)$ denotes a score function representing the interaction of $x$ with respect to $\P_n$. When $\xi$ depends on local data in the sense that its radius of stabilization is well controlled, we establish expectation asymptotics, variance asymptotics, and central limit theorems for $H_n^\xi$ as well as for the random measures $\sum_{x \in \P_n} \xi(x, \P_n) \delta_{n^{-1/d} x},$ as $W_n \uparrow \R^d$. This gives the limit theory for non-linear geometric statistics of determinantal point processes with fast decreasing kernels, including the Ginibre ensemble, extending the Gaussian fluctuation results of Soshnikov to non-linear statistics. It also gives limit theory for geometric statistics of permanental input as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nararov and Sodin, which are also confined to linear statistics. In this way we obtain limit theory for statistics of simplicial complexes, Morse critical points, germ-grain models, and random graphs whenever the input is clustering. Our approach depends on a factorial moment expansion introducef by Blaszczyszyn in 1995 for expected values of functions of general point processes.

czwartek, 15-10-2015 - 12:15, 602
Branching Random Walks, Stable Point Processes and a Conjecture of Brunet and Derrida.
Ayan Bhattacharya

Stable point processes were introduced and characterized by Davydov, Molchanov and Zuyev (2008). They showed that such point processes can always be represented as a scale mixture of iid copies of one point process with the scaling points coming from an independent Poisson random measure. We obtain such a point process as a weak limit of a sequence of point processes induced by a branching random walk with regularly varying displacements. In particular, we show that a conjecture of two physicists, Brunet and Derrida (2011), remains valid in this setup, and recover a result of Durrett (1983). This talk is based on a joint work with Rajat Subhra Hazra and Parthanil Roy.

czwartek, 14-05-2015 - 12:15, 602
On optimal dividend problem for an insurance risk models with surplus-dependent premiums
Zbigniew Palmowski

In this talk we present an optimal dividend distribution problem for an insurance company with surplus-dependent premium. In the absence of dividend payments, such a risk process is a particular case of so-called piecewise deterministic Markov processes. The control mechanism chooses the size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividend payments received until the time of ruin and a penalty payment at the time of ruin which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. We identify the associated Hamilton-Jacobi-Bellman equation and nd necessary and su cient conditions for optimality of a single dividend-band strategy, in terms of particular Gerber-Shiu functions. A number of concrete examples are analyzed.