The Eighth Congress of Romanian Mathematicians

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List of talks

I. Algebra and Number Theory

Special session: Local rings and homological algebra. Special session dedicated to Prof. Nicolae Radu

II. Algebraic, Complex and Differential Geometry and Topology

Special session: Geometry and Topology of Differentiable Manifolds and Algebraic Varieties

III. Real and Complex Analysis, Potential Theory

IV. Ordinary and Partial Differential Equations, Variational Methods, Optimal Control

Special session: Optimization and Games Theory

V. Functional Analysis, Operator Theory and Operator Algebras, Mathematical Physics

Special session: Spectral Theory and Applications in Mathematical Physics

Special session: Dynamical Systems and Ergodic Theory

VI. Probability, Stochastic Analysis, and Mathematical Statistics

VII. Mechanics, Numerical Analysis, Mathematical Models in Sciences

Special session: Mathematical Modeling of Some Medical and Biological Processes

Special session: Mathematical Models in Astronomy

VIII. Theoretical Computer Science, Operations Research and Mathematical Programming

Special session: Logic in Computer Science

IX. History and Philosophy of Mathematics

Probability, Stochastic Analysis, and Mathematical Statistics 

(this list is in updating process)

ANTON Cristina
Grant MacEwan University, Canada
Title: Statistical Analysis of a Cytotoxicity Model (details)
Recently there is a big interest to develop innovative toxicity profiling programs based on large data sets obtained through experimental studies. In this work we develop a mathematical model representing the effect of chemical compounds on the growth/death of different kinds of cells. This model is represented by a non-linear state-space model. To estimate the parameters of the model we use the expectation maximization (EM) algorithm. Since the state equation is non-linear, an approximation is needed during the E-step. The likelihood and the conditional likelihood are approximated based on a linearization, and the unscented filter is used for filtering, smoothing and prediction. The model is validated using experimental cytotoxicity data.
Universite Paris Est Marne la Vallée, France
Title: Asymptotic behavior for PDMP's with three regime (details)
We consider a sequence of Picewise Deterministic Markov Process (PDMP) with three regimes: a rapid one (of CLT type), a medium one (of Law of Large Numbers type) and a slow one and we study the asymptotic behavior of such a sequence. At the limit the rapid regime gives rise to a diffusion component, the medium one to a drift component and the slaw one to a finite variation jump process. So the limit equation is no more a PDMP but an equation with diffusive behavior between the jump times and with an infinity of jumps in each finite time interval. This type of equation seems new in the literature. We prove existence and uniqueness for it and we study the regularity of the semigroup. Finally we prove the convergence result and we obtain estimates of the rate of convergence.
BARBU Vlad - Stefan
Universite de Rouen, Laboratoire de Mathematiques Raphael Salem, France
Title: Survival analysis for semi-Markov systems (details)
Abstract Semi-Markov processes and Markov renewal processes represent a class of stochastic processes that generalize Markov and renewal processes. As it is well known, for a discrete-time (respectively continuous-time) Markov process, the sojourn time in each state is geometrically (respectively exponentially) distributed. In the semi-Markov case, the sojourn time distribution can be any distribution on (respectively on ). This is the reason why the semi-Markov approach is much more suitable for applications than the Markov one. The purpose of our talk is threefold: (i) to make a general introduction to semi-Markov processes; (ii) to investigate some survival analysis and reliability problems for this type of system and (iii) to address some statistical topics. We start by briefly introducing the discrete-time semi-Markov framework, giving some basic definitions and results. These results are applied in order to obtain closed forms for some survival or reliability indicators, like survival/reliability function, availability, mean hitting times, etc; we also discuss the particularity of working in discrete time. The last part of our talk is devoted to the nonparametric estimation of the main characteristics of a semi-Markov system (semi-Markov kernel, semi-Markov transition probabilities, etc) and to the asymptotic properties of these estimators. Statistical issues for the reliability indicators are also presented. Bibliography V. Barbu, N. Limnios, Some algebraic methods in semi-Markov processes, In Algebraic Methods in Statistics and Probability, volume 2, series Contemporary Mathematics edited by AMS, Urbana, 19-35, 2010. V. Barbu, N. Limnios, Semi-Markov Chains and Hidden Semi-Markov Models toward Applications - Their use in Reliability and DNA Analysis, Lecture Notes in Statistics, vol. 191, Springer, New York, 2008. V. Barbu, N. Limnios, Reliability of semi-Markov systems in discrete time: modeling and estimation, In Handbook on Performability Engineering (ed. K. B. Misra), Springer, 369-380, 2008.
University Politehnica of Bucharest, Romania
Title: Modeling and calibrating banks' demand deposits versus asset sizes (details)
We model and calibrate U.S. banks' demand deposits and we study the relationship between banks' asset sizes and the estimated parameters of the deposit processes. Assuming that the banks' demand deposits evolve as a Brownian motion (geometric Brownian motion/ Ornstein-Uhlenbeck process/ geometric Ornstein- Uhlenbeck process, respectively) between March 1991 and December 2000, we estimate the corresponding drifts and volatilities. The goodness-of-fit tests show that the best model among the proposed ones is the geometric Ornstein-Uhlenbeck process, followed by the Ornstein-Uhlenbeck process, the geometric Brownian motion and the Brownian motion with drift, respectively. Regarding the connection between the asset sizes and deposits, we give the parameters of the deposits as functions of the mean asset sizes, using a regression line.
Simion Stoilow Institute of Mathematics of the Romanian Academy, Romania
Title: A new approach to the existence of invariant measures for Markovian semigroups (details)
We present a result concerning the existence of invariant measures for a Markovian semigroup $(P_t)_t$" consisting of two steps: first we identify a convenient auxiliary measure $m$ and then we study the existence of a non-zero co-excessive function for $(P_t)_t$ regarded as a semigroup on $L^\infty (m)$. As an application we provide short proofs for the theorems of Lasota and Harris, and we answer to an open problem mentioned by Tweedie, concerning the sufficiency of the generalized drift condition for the existence of an invariant measure. We improve several results on the existence of invariant measures for small perturbations of Dirichlet forms, due to V. Bogachev, M. R\"ockner, and T.S. Zhang.
CIUIU Daniel
Technical University of Civil Engineering Bucharest; Romanian Institute for Economic Forecasting, Romania
Title: Bayesian good-of-fit tests: past, present and future (details)
\documentclass[12pt,a4paper,twoside]{article} \usepackage{amsfonts} \usepackage{amstext} \usepackage{amsmath} \usepackage{amscd} \usepackage{amssymb} \usepackage{graphics} \usepackage{graphicx} \usepackage{float} \usepackage{indentfirst} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}{Definition} \newtheorem{remark}{Remark} \newtheorem{example}{Example} \newenvironment{proof}{\begin{quote}{\bf Proof:}}{$\Box$ \end{quote}} \addtolength{\oddsidemargin}{-5mm} \addtolength{\evensidemargin}{-2cm} \addtolength{\topmargin}{-2cm} \addtolength{\textwidth}{2.5cm} \addtolength{\textheight}{3.5cm} \begin{document} \title{Bayesian good-of-fit tests: past, present and future} \author{Daniel Ciuiu\\ Technical University of Civil Engineering Bucharest;\\ Romanian Institute for Economic Forecasting\\ e-mail:\\ and\\ Carlos Mat\' e\\ Universidad Pontificia Comillas Madrid\\ e-mail: } \date{} \maketitle \begin{abstract} In this paper we will build the Bayesian version for the good-of-fit tests $\chi^2$ and Kolmogorov---Smirnov. Because for % the last test the theoretical distribution must be totally specified, we will divide first the sample in two parts: % the first part is for inference, and the second part is for test. The completely specified theoretical cdf for the % second part of the sample is the Bayesian forecasted cdf from the first part. This is unique if the prior distribution % is fixed. For the $\chi^2$ test, we do the same Bayesian inference in the first part, and we perform % the Bayesian forecasts for the probability such that $X$ belongs to the involved intervals (the values of $p_i$). % The parameters of the prior distribution are chosen such that the $\chi^2$ statistics is minimum, % and the number of degrees of freedom is $k-1-npar$, where $k$ is the number of intervals, and $npar$ % is the number of parameters of $X$. Of course, we can fix the prior distribution as for % Kolmogorov---Smirnov test, but the number of degrees of freedom is $k-1$. For the last test we can consider the whole sample, and the parameters that characterise the distribution % of $X$ are the Bayesian estimators. The number of degrees of freedom are the same as above, and % $npar$ is again the number of parameters of the distribution of $X$. When we estimate the values of forecasted cdf/ forecasted probabilities of the intervals % or when we estimate the parameters for the chi square test we apply analytical formulae if they exist. Otherwise, % we generate a sample according the forecasted distribution of $\left.X\right|S$ (or the posterior distribution of $\left.\theta\right|S$), and % next we apply the Monte Carlo method. The way we % generate the values of $X$ is to use the mixture method: we generate $\theta$ according the posterior distribution, % and $X$ is generated for each $\theta$.\\ \textbf{Keywords}: Bayesian forecasting, Bayesian estimators, good-of-fit tests.\\ \textbf{AMS 2010 Subject Classification}: 62F15, 62C12, 62H15. \end{abstract} \end{document}
Bioclinome, France
Title: Voiculescu's free entropy and spectral analysis of random graphs (details)
Using the free probability calculus initiated by Dan Voiculescu we study the spectral measures of different random graphs. Some examples from network communications and computational physiology are presented.
Case Western Reserve University, U.S.A.
Title: Stochastic aspects of Single Cell Analysis (details)
New technologies in molecular biology have made it possible, in the last few years, to accurately measure the levels of gene activity, of epigenetic modifications, and even the sequencing of whole genomes, using the nuclear material of a single cell. At this level of detail, statistical analysis needs to take into account not only measurement error but also stochastic variation from cell to cell in order to obtain sensible inferences. We will survey some of the work done in this cutting-edge field of research.
DEACONU Madalina
Inria Centre de Recherche - Nancy Grand-Est, France
Title: Brownian and Bessel hitting times: new trends in their approximation (details)
A new method for the simulation of the exit time and position, of the $\delta$-dimensional Brownian motion and general Bessel processes, from a domain is constructed. This method avoids splitting time schemes as well as inversion of complicated series. We introduce first the walk on moving spheres algorithm for approximating hitting times of Bessel processes with integer dimension. This new method couples the method of images for the first hitting time, of a non-linear boundary for the Brownian motion, and the random walk on the spheres method, for the heat equation. After, the hitting time of a non-integer Bessel process is approximated by using the additivity property of the distributions of squared Bessel processes. Each simulation step is split in two parts~: one is using the integer dimension case and the other one considers hitting times for a Bessel process starting from zero. By using the connexion between the $\delta$-dimensional Bessel process and the $\delta$-dimensional Brownian motion we construct a fast and accurate numerical scheme for approximating the exit time and position from a boundary for the $\delta$-dimensional Brownian motion. This is a joint work with Samuel Herrmann (University of Burgundy, France) and Sylvain Maire (University of Toulon, France).
Universite Paris-Est, France
Title: Asymptotic Control of Switch Processes in Systems Biology (details)
We begin by recalling the construction of a class of hybrid stochastic processes (piecewise deterministic/ diffusive Markov processes). This is done in connection to stochastic gene networks (e.g. moderate viruses). We discuss nonexpansive conditions guaranteeing the existence of long-run averaged value functions and generalized Abel/Tauberian results.
Universite de Rennes 1, France
Title: Nonlinear Langevin type equation driven by stable Lévy process (details)
The speed $v^epsilon_t$ of a particle is a solution of a sde driven by a small $alpha$-Levy process $epsilon\times l_t$ with non-linear drift coefficient $-sgn(v)|v|^beta$, $beta>2-alpha/2$. One studies the asymptotics of the position $x^epsilon _t$ of the particle, under appropriate normalization, when $epsilon$ tends to 0 and different limits are emphasized. The talk is based on joint works with Ilya Pavlyukevich (Jena) and with Richard Eon (Rennes).
HSU Elton P
Northwestern University and University of Science and Technology of China, China
Title: Brownian Motion on Complex Structures (details)
There is a well developed theory of Brownian motion on Riemannian manifolds. In this talk, we will study properties of Brownian motion on a complex domain or a complex manifold under various Riemannian metrics related to the complex structure of the underlying spaces. In particular, we will clarify probabilistic meanings of the Kaehler property and pseudo convexity and show how they affect the behavior of Brownian motion on complex structures with these properties. The cases of the unit ball and a polydisc can be studied by explicit computations.
Univ. of Gottingen, Germany
Title: On Relations Between Statistics and Geometry (details)
Multivariate statistics takes great advantage from linearization due to the linear structure of the data space. If data reside in non-linear spaces, which, for instance, occurs in shape analysis, already the most basic statistical concepts such as means or principal components can no longer be simply defined as arithmetic averages or eigenspaces of covariance matrices. For suitable data descriptors living on manifolds or stratified spaces we investigate asymptotic properties to allow for statistical inference. It turns out that rates and forms of central limit theorems may reflect the topological and geometric structure of the underlying space.
LAZARI Alexandru
Moldova State University, Republic of Moldova
Title: Geometric Programming Models for Dynamical Decision Stochastic Systems with Final Sequence of States (details)
This paper describes several classes of dynamical stochastic systems, that represent an extension of classical Markov decision processes. The Markov stochastic systems with given final sequence of states, over a finite or infinite state space, are studied. Such dynamical system stops its evolution as soon as given sequence of states in given order is reached. The evolution time of the stochastic system with fixed final sequence of states depends on initial distribution of the states and probability transition matrix. We are seeking for the optimal initial distribution and optimal probability transition matrix, that provide the minimal evolution time for the dynamical system. We show that this problem can be solved using the geometric programming approach. These geometric programming models are developed and theoretically grounded.
Université de Cergy-Pontoise, France
Title: Propagation of chaos for systems of interacting neurons (details)
We study the hydrodynamic limit of a stochastic process describing the time evolution of a system with $N$ neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value $0$ and, simultaneously, the membrane potentials of the other neurons are increased by an amount of {sl energy} $frac{1}{N} $. This mimics the effect of chemical synapses. Additionally, the effect of electrical synapses is represented by a deterministic drift of all the membrane potentials towards the average value of the system. We show that, as the system size $N$ diverges, the distribution of membrane potentials becomes deterministic and is described by a limit density which obeys a non linear PDE which is a conservation law of hyperbolic type.
Institute of Mathematical Statistics and Applied Mathematics and Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, and the Research Institute of the University of Bucharest, Romania
Title: Branching processes and the fragmentation equation (details)
We investigate branching properties of the solution of a stochastic differential equation of fragmentation and we properly associate a continuous time cadlag Markov process on the space of all fragmentation sizes, introduced by J. Bertoin. The construction and the proof of the path regularity of the Markov processes are based on several newly developed potential theoretical tools, in terms of excessive functions and measures, compact Lyapunov functions, and some appropriate absorbing sets. The talk is based on joint works with Lucian Beznea and Madalina Deaconu.
University of North Carolina, U. S. A.
Title: Object Oriented Data Analysis (details)
Object Oriented Data Analysis is the statistical analysis of populations of complex objects. In the special case of Functional Data Analysis, these data objects are curves, where standard Euclidean approaches, such as principal components analysis, have been very successful. In non-Euclidean analysis, the approach of Backwards PCA is seen to be quite useful. An overview of insightful mathematical statistics for object data is given
"Gheorghe Asachi" Technical University of Iasi, Romania
Title: Viscosity solutions for functional parabolic PDEs. A stochastic approach via BSDEs with time-delayed (details)
We provide a probabilistic representation of a viscosity solution for the path dependent nonlinear Kolmogorov equation: \begin{equation} \left\{ \begin{array}{r} -\partial _{t}u(t,\phi )-\mathcal{L}u(t,\phi )-f(t,\phi ,u(t,\phi ),\partial _{x}u(t,\phi )\sigma (t,\phi ),(u(\cdot ,\phi ))_{t})=0,\medskip \\ \multicolumn{1}{l}{u(T,\phi )=h(\phi ),\;t\in \lbrack 0,T),\;\phi \in \mathcal{C}([0,T];\mathbb{R}^{d}),}% \end{array}% \right. \label{1} \end{equation}% where $\mathcal{L}$ is a second order differential operator and $(u(\cdot ,\phi ))_{t}:=(u(t+\theta ,\phi ))_{\theta \in \lbrack -\delta ,0]}\,$. We show that the functional $u:[0,T]\times \mathcal{C}([0,T];\mathbb{R}% ^{d})\rightarrow \mathbb{R}$, $u(t,\phi):=Y^{t,\phi}(t)$, is a viscosity solution for (1), where $(Y^{t,\phi },Z^{t,\phi })$ is the unique solution of a BSDE with time delayed--generators.
Georgia Institute of Technology, Atlanta, United States
Title: Sample Size Needed for Estimating Principal Component (details)
We show how we can get a detailed description of the estimation error of certain eigenvectors of a covariance matrix even in the high-dimensional case. This allows in may cases to get smaller bounds for the sample size needed to retrieve these eigenvectors precisely.
Kent State University, U.S.A.
Title: Stochastic modeling of compositional data with diffusions (details)
Here we present a class of stochastic processes in continuous time which take as values vectors with non-negative values adding up to 1, and show their use as models for continuously changing compositions. They are defined as solutions of a stochastic differential equation, in such a way that the marginal distributions are Dirichlet. The aggregation property of the Dirichlet distribution is exploited to allow the study of microbial compositions at different taxonomic levels in microbiome surveys.
NOVAC Ludmila
Moldova State University, Republic of Moldova
Title: Approach of the Currency Exchange Risk (details)
In this article we analyze the problem of Currency Exchange. The main goal is to approach the risk of loss for this problem (so called Currency Exchange Risk). Currency changes affect us, whether we are actively trading in the foreign exchange market, planning our next vacation, shopping online for goods from another country or just buying food or other things imported from abroad. More exactly, we can define this risk (also known as ”currency risk” or ”exchange-rate risk”) as the risk that an investor will have to close out a long or short position in a foreign currency at a loss due to an adverse movement in exchange rates. In order to analyze some multistage interactive situations of this problem, we will use the theory of extensive games and we will construct a dinamical model. Every such a situation can be represented by a strategic game. Representing a model as a multistage interactive situation, we can construct a dynamical model for the problem of the Currency Exchange. This risk usually affects businesses that export and/or import, but it can also affect investors making international investments. For example, if money must be converted to another currency to make a certain investment, then any changes in the currency exchange rate will cause that investment’s value to either decrease or increase when the investment is sold and converted back into the original currency. There are many factors that could cause the fluctuation of the currency rate. The supply and demand of a country’s money is reflected in its foreign exchange rate. It is not simple to determine the risk of the not adequate business. Consumer spending is influenced by a number of factors: the price of goods and services (inflation), employment, interest rates, government initiatives, and so on. There are many economic factors we can follow to identify economic trends and their effect on currencies. Keywords: dynamical model, problem of currency exchange, minimum loss, profit, loss risk, currency exchange risk, currency risk, exchange-rate risk.
PASCU Mihai N.
Transilvania University of Brasov, Faculty of Mathematics and Computer Science, Romania
Title: Brownian Couplings and Applications (details)
\documentclass{article} \usepackage[margin=2cm]{geometry} \usepackage{hyperref} \title{Brownian Couplings and Applications} \date{\it{Transilvania} University of Bra\c{s}ov, Faculty of Mathematics and Computer Science, Str. Iuliu Maniu 50, Bra\c{s}ov -- 500091, Romania. \\Email:} \begin{document} \maketitle Brownian motion is invariant under the three basic geometric transformation: translation, scaling and rotation/symmetry. In this talk we will show that corresponding to each of these invariance properties one can construct couplings of (reflecting) Brownian motions: \emph{scaling coupling} (\cite{PascuScaling}), \emph{mirror coupling} (\cite{AtarBurdzy1}, \cite{AtarBurdzy2}, \cite{PascuMirror}), and recently the \emph{``translation'' coupling} (\cite{PascuPopescu}). Aside from the intrinsic interest of the construction (existence of a pair of RBM with specified properties), the existence and the properties of these couplings can be used to prove various results for certain functionals associated with the RBM. For example, as an application of the scaling coupling, we will prove a monotonicity of the lifetime of reflecting Brownian motion with killing, which implies the validity of the \emph{Hot Spots conjecture} of J. Rauch for a certain class of domains. As applications of the mirror coupling, we will present the proof of the \emph{Laugesen-Morpurgo conjecture} (\cite{PascuLaugesen}), and a unifying proof of the results of I. Chavel and W. Kendall on \emph{Chavel's conjecture}. Time-permitting, we will discuss the recent results (joint with I. Popescu, \cite{PascuPopescu}) on fixed-distance couplings, a particular case of \emph{shy coupling} (as introduced and studied in \cite{Burdzy-Benjamini},\cite{Burdzy-Kendall}, \cite{Kendall}), and some of its applications. For example, in the particular case of a sphere, the existence of this coupling gives a resolution for a stochastic version of the celebrated \emph{Lion and Man problem} of Rad\'{o} (\cite{Littlewood}). %{\footnotesize{ \begin{thebibliography}{1} \bibitem{AtarBurdzy1} R.~Atar, K.~Burdzy, \emph{On Neumann eigenfunction in Lip domains}, JAMS, \textbf{17 }(2004), No. 2, pp. 243 -- 265. \bibitem{AtarBurdzy2} R. Atar, K. Burdzy, \emph{Mirror couplings and Neumann eigenfunctions}, Indiana Univ. Math. J., \textbf{57 }(2008), pp. 1317 -- 1351. \bibitem{Burdzy-Benjamini}I.~Benjamini, K.~Burdzy, Z.Q.~Chen, \emph{Shy couplings}, Probab. Theory Related Fields \textbf{137} (2007), No. 3-4, pp. 345 -- 377. \bibitem{Burdzy-Kendall}M.~Bramson, K.~Burdzy, W.S.~Kendall, \emph{Shy couplings, {CAT(0)} spaces, and the lion and man}, Ann. Probab. \textbf{41} (2013), No. 2, pp. 744 -- 784. \bibitem{Kendall}W.S.~Kendall, \emph{Brownian couplings, convexity, and shy-ness}, Electron. Commun. Probab. \textbf{14} (2009), pp. 66 -- 80. \bibitem{Littlewood} J.E.~Littlewood, \emph{Littlewood's miscellany}, Cambridge University Press, Cambridge, 1986, Edited and with a foreword by B{\'e}la Bollob{\'a}s. \bibitem{PascuScaling} M.N.~Pascu, \emph{Scaling coupling of reflecting Brownian motions and the hot spots problem}, Trans. Amer. Math. Soc. \textbf{354} (2002), No. 11, pp. 4681 -- 4702. \bibitem{PascuLaugesen} M.N.~Pascu, \emph{Monotonicity properties of Neumann heat kernel in the ball}, J. Funct. Anal. \textbf{260} (2011), No. 2, pp. 490 -- 500. \bibitem{PascuMirror} M.N.~Pascu, \emph{Mirror coupling of reflecting Brownian motion and an application to Chavel's conjecture}, Electron. J. Probab. \textbf{16} (2011), No. 18, pp. 504 -- 530. \bibitem{PascuPopescu} M.N.~Pascu, I.~Popescu, \emph{Shy and Fixed-Distance Couplings of Brownian Motions on Manifolds}, preprint (\href{}{Arxiv link}). \end{thebibliography}
Florida State University, U S A
Title: Two Sample Tests for Means on Lie Groups and Homogeneous Spaces with Examples (details)
This presentation focuses on recent advances in Data Analysis on Homogeneous Spaces, including two sample tests and MANOVA for means of random objects on homogeneous spaces with examples from 3D high level image analysis
PIRVU Traian
McMaster University, Canada
Title: Cumulative Prospect Theory with Skewed Return Distribution (details)
We investigate a one-period portfolio optimization problem of a cumulative prospect theory (CPT) investor with multiple risky assets and one risk-free asset. The returns of multiple risky assets follow multivariate generalized hyperbolic (GH) skewed t distribution. We obtain a three-fund separation result of two risky portfolios and risk-free asset. Furthermore, we reduce the high dimensional optimization problem to two 1-dimensional optimization problems and derive the optimal portfolio. We show that the optimal portfolio composition changes as some of investor-specific parameters change. It is observed that the consideration of skewness of stock return distribution has considerable impact on the distribution of CPT investor's wealth deviation, and leads to less total risky investment.
ROBE - VOINEA Elena - Gratiela
University of Bucharest, Romania
Title: On the recursive evaluation of a certain multivariate compound distribution (details)
Abstract In this paper, we extend to a multivariate setting the bivariate model introduced by Jin and Ren (2014) to model insurance aggregate claims in the case when different types of claims simultaneously affect an insurance portfolio. We obtain an exact recursive formula for the probability function of the multivariate compound distribution corresponding to the model A of Jin and Ren (2014), under the assumption that the conditional multivariate counting distribution (conditioned by the total number of claims) is multinomial. Our formula extends the corresponding one from Jin and Ren (2014). Key words: insurance model, multivariate aggregate claims, multinomial distribution, recursion.
Bielefeld University, Germany
Title: A new approach to stochastic PDE (details)
In this paper, we develop a new general approach to the existence and uniqueness theory of infinite dimensional stochastic equations of the form $dX+A(t)X dt=X dW$ in $(0,T)\times H$, where $A(t)$ is a nonlinear monotone and demicontinuous operator from $V$ to $V'$, coercive and with polynomial growth. Here, $V$ is a reflexive Banach space continuously and densely embedded in a Hilbert space $H$ of (generalized) functions on a domain $\mathcal{O}\subset\mathbb{R}^d$ and $V'$ is the dual of $V$ in the duality induced by $H$ as pivot space. Furthermore, $W$ is a Wiener process in $H$. The new approach is based on an operatorial reformulation of the stochastic equation which is quite robust under perturbation of $A(t)$. This leads to new existence and uniqueness results of a larger class of equations with linear multiplicative noise than the one treatable by the known approaches. In addition, we obtain regularity results for the solutions with respect to both the time and spatial variable which are sharper than the classical ones. New applications include stochastic partial differential equations, as e.g. stochastic transport equations.
Alexandru Ioan Cuza University of Iasi, Romania
Title: Anticipated BSVIs with generalized reflection (details)
We present existence and uniqueness results for anticipated backward stochastic variational inequalities with generalized reflecting directions. In these equations the generator includes not only the values of solutions of the present, but also the future. Numerical approximation schemes are also envisaged.
Babes Bolyai University, Romania
Title: Stochastic spline fractal interpolation functions (details)
The spline interpolation method is the most important and well-known classical real data interpolation method. It has a lot of applications especially in computer geometric design. But the classical method can be generalized with fractal interpolation. These fractal interpolation functions provide new methods of approximation of exprimental data. This paper extends the spline fractal interpolation method to stochastic case.
TONE Cristina
University of Louisville, USA
Title: Functional Central Limit Theorem for Empirical Processes under a Strong Mixing Condition (details)
We introduce a functional central limit theorem for empirical processes endowed with real values from a strictly stationary random field satisfying an interlaced mixing condition. We proceed by first obtaining the limit theorem for the uniformly distributed case. We then generalize the result to the case where the absolutely continuous marginal distribution function is no longer uniform. In this case we show that the empirical process endowed with values from the rho′-mixing stationary random field, due to the strong mixing condition, doesn't converge in distribution to a Brownian bridge,but to a continuous Gaussian process with mean zero and the covariance given by the limit of the covariance of the empirical process.
Seoul National University, South Korea
Title: Recurrence criteria for diffusion processes generated by divergence free perturbations of non-symmetric energy forms (details)
On a metric space $E$, we consider a generalized Dirichlet form \[ \mathcal{E}(f,g)=\mathcal{E}^0(f,g)+\int_E f Ng\, d\mu, \] where $(\mathcal{E}^0,D(\mathcal{E}^0))$ is a sectorial Dirichlet form on $L^2(E,d\mu)$, $(N,D(N))$ is a linear operator on $L^2(E,d\mu)$ and $\mathcal{E}^0(f,f)\le \mathcal{E}(f,f)$. We find a criterion for recurrence of $\mathcal{E}$. Namely, if the generalized Dirichlet form $\mathcal{E}$ is strictly irreducible and if there exists a sequence of functions $(\chi_n)_{n\ge 1}$ with $0\le \chi_n\le 1$, $\lim_{n\to \infty}\chi_n=1$ $\mu$-a.e. satisfying \begin{equation}\label{1} \lim_{n\to \infty}\left (\mathcal{E}^0(g,\chi_n)+\int_E g N\chi_n\, d\mu\right )=0 \end{equation} for any non-negative bounded $g$ in the extended Dirichlet space of $D(\mathcal{E}^0)$, then $\mathcal{E}$ is recurrent. As application, we consider $E\subset \mathbb{R}^d$, $E$ open or closed and a strictly irreducible generalized Dirichlet form \[ \mathcal{E}(f,g)=\frac12\int_E A(\nabla f)\cdot \nabla g\, d\mu-\int_E (B\cdot \nabla f) g\, d\mu, \] where the diffusion matrix $A=(a_{ij})$ is not necessarily symmetric but its antisymmetric part consists of bounded functions and $B$ is a locally $\mu$-square integrable $\mu$-divergence free vector field. Then using volume growth conditions of $B$ and of the sectorial part of $\mathcal{E}$ on Euclidean balls w.r.t. $\mu$, we construct explicitly $(\chi_n)_{n\ge 1}$ satisfying (\ref{1}). One astonishing observation is that there may exist a sequence of functions $(\chi_n)_{n\ge 1}$ in the Dirichlet space with $0\le \chi_n\le 1$, $\lim_{n\to \infty}\chi_n=1$ $\mu$-a.e. and such that $\lim_{n\to \infty}\mathcal{E}(\chi_n,\chi_n)=0$, but recurrence does not hold for $\mathcal{E}$ in contrast to the symmetric case. Finally, as a concrete example, we consider a whole class of singular diffusions associated to $\mathcal{E}$, where $\mu=\rho\, dx$, $\rho$ is continuous and in a certain Muckenhoupt class. Here, we discuss non-explosion (in particular existence of a pathwise and weakly unique strong solution) and recurrence for any initial condition $x\in \{\rho>0\}$.
Constantin Brancusi University, Romania
Title: Stabilizing solution for modified algebraic Riccati equations in infinite dimensions (details)
The study of optimal control problems associated to stochastic systems with Markovian regime switching are of particular interest to researchers due to their various applications in finance, biology, engineering etc. In particular, infinite horizon linear quadratic (LQ) optimal control problems for stochastic systems with Markovian jumps has been recently investigated for both finite and infinite-dimensional cases. Their solvability is closely related to the existence of mean-square stabilizing (MSS) solutions for a class of modified Riccati equations (MREs). Most existing works, prove the existence of MSS solutions for these MAREs under either detectability or observability conditions. Some others give necessary and sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). In this talk we consider a class of infinite-dimensional algebraic MREs associated with time-invariant linear stochastic systems affected simultaneous by multiplicative white noise(MN) and Markovian jumps(MJs). These algebraic MREs are defined on ordered Banach spaces of sequences of linear and bounded operators. Using spectral properties of positive operators and the general theory of compact (in particular, of trace-class operators), we provide necessary and sufficient conditions for the existence of MSS solutions for infinite-dimensional algebraic MREs in terms of system parameters. and of trace class and compact operators. mean-square stabilizing solution in terms of stabilizability and observability. Our results generalize similar ones obtained for finite-dimensional MREs associated with stochastic linear systems without MJs.
Educational Testing Service, USA
Title: Psychometric Applications: Parameter Estimation and Comparability of Test Performance in Multistage Testing (details)
In a Multistage adaptive Test (MST), all test takers take first a set of items (called the routing module), and then, based on their performance there are administered different sets of items (or modules). Parameter estimation, calibration, linking, scoring, and equating are statistical processes required for any practical implementation of an MST design. This presentation discusses the types of parameter estimation and linking necessary at different stages in the life of an operational MST program. Item response theory (IRT) has traditionally provided the framework for these processes in analyzing test data. Justification is provided for application of conventional estimation methods for IRT to an MST. Calibration and linking involve an initial phase for data collection in which conventional test administrations are used to build modules and routing rules to begin use of MST administrations. Once MST administrations begin, data are collected to establish scoring rules, develop new test modules, equate the cut-scores for routing, and ensure comparability of tests over time. Discussion involves both direct use of estimated examinee proficiency in routing and scoring and use of sum scores for these purposes. When many administrations are involved, special procedures are considered for maintenance of stable linkage