Sztochasztikus modellek statisztikai vizsgálata = Statistical investigation of stochastic models

A regressziós modellek témakörében bevezettünk egy új becslést, igazoltuk annak konzisztenciáját és aszimptotikus normalitását. Különböző térbeli autoregresszív folyamatok és egy időbeli SAR folyamat esetén leírtuk a paraméterek legkisebb négyezetes becslésének aszimptotikus tulajdonságait. Meghatároztuk a Wiener mező eltolásparaméterének maximum likelihood (ML) becslését. Új eredményeket értünk el lokálisan kompakt topológikus csoportokon, Lie-csoportokon értelmezett valószínűségi mértékek analitikus és algebrai tulajdonságainak meghatározása terén. Időhomogén diffúziós folyamatok egy osztályára meghatároztuk bizonyos paraméterek ML becslésének határeloszlását, megadtuk a folyamat egyes funkcionáljainak Laplace transzformáltját és vizsgáltuk az alpha-Wiener hidak trajektóriáinak regularitási tulajdonságait. Speciális diszkrét idejű forward kamatlábmodellek esetén igazoltuk a paraméterek ML becslésének konzisztenciáját és aszimptotikus normalitását. A javasolt arbitrázsmentes forward kamatlábmodellekben foglalkoztunk modellszelekciós kérdésekkel is. Olyan likelihood hányados próbákat készítettünk, amelyek alapján lehetővé válik a modellek összevetése. Vizsgáltuk a Lee-Carter módszert és változatait, és ezek alapján előrejelzéseket adtunk a magyar halandósági ráták alakulására. Teszteltük a Wilkie modellt, alternatív modelleket adtunk meg, melyek segítségével aktuárius alkalmazások céljából elkészítettük néhány magyar makrogazdasági mutató előrejelzését. | In the field of regression models we introduced a new estimator and we proved it's consistency and asymptotic normality. For various spatial autoregressive processes and for a SAR model on integers we described the asymptotic properties of the least squares estimators of the parameters. We determined the maximum likelihood (ML) estimator of the shift parameter of a Wiener sheet. We obtained new results in describing the analytic and algebraic properties of probability measures defined on locally compact topological groups, Lie-groups. For some time homogeneous diffusion processes we determined the limiting distributions of the ML estimators of certain parameters, calculated the Laplace transforms of some functionals of the process and we also investigated the regularity properties of trajectories of alpha-Wiener bridges. For special discrete time forward interest rate models we proved the consistency and asymptotic normality of the ML estimators of the parameters. We also delt with model selection problems in the proposed arbitrage free forward rate models. We described likelihood ratio tests that make possible the comparison of the models. We investigated the Lee-Carter method and modified versions, and based on these models we gave predictions of the Hungarian mortality rates. Finally, we tested the so-called Wilkie model, we gave alternative models and for actuarial purposes we calculated the predictions of some Hungarian macroeconomic quantities

Határeloszlástételek és alkalmazásaik = Limit theorems with applications

Kidolgoztunk egy Heath-Jarrow-Morton típusú diszkrét idejű határidős kamatlábmodellt, melyet egy autoregressziós mező hajt meg, nem pedig egyetlen folyamat, mely realisztikusabb, mint az eredeti modell. Drift-feltételt vezettünk le az arbitrázsmentességre, és különböző statisztikai kérdéseket vizsgáltunk meg, többek között konzisztenciát, valamint a paraméterek becslésének aszimptotikus viselkedését mind stabil, mind pedig instabil esetekben. Sikerült Black-Scholes formulát levezetni késleltetett modellekben is. Levezettünk elégséges feltételeket valószínűségi változókból álló háromszögrendszerre, melynek teljesülése esetén a háromszögrendszerből felépített véletlen lépcsősfüggvények egy (nem feltétlenül időhomogén) diffúziós folyamathoz konvergálnak. Továbbá elégséges feltételeket adtunk arra, hogy véletlen lépcsősfüggvények sztochasztikus integréljainak sorozata konvergáljon egy sztochasztikus integrálhoz, amikor az integrandusok az integrátorok funkcionáljai. Különböző eredményeket értünk el inhomogén diffúziós folyamatok statisztikai kérdéseivel kapcsolatban. Új eredményeink vannak térbeli folyamatok statisztikai viselkedésével kapcsolatban is, mind stabil, mind pedig instabil esetekben, mind diszkrét, mind folytonosidőben. Egzakt formulát kaptunk Heisenberg-csoporton értelmezett Gauss-mérékek Fourier-transzformáltjaira. Új központi határelsozlás-tételeket kaptunk lokálisan kompakt Abel-csoportokon. | We worked out a discrete time Heath-Jarrow-Morton type interest rate model driven by an autoregressive random field instead of a single process, which is more realistic than the original one. We derived no-arbitrage drift-condition, and investigated several statistical questions, including consistency and asymptotic behavior of maximum likelihood estimator of the parameters both in stable and unstable cases. We also derived a delayed Black-Scholes formula. We derived sufficient conditions for a triangular array of random vectors such that the sequence of related random step functions converges towards a (not necessarily time homogeneous) diffusion process. Sufficient conditions are also given for convergence of stochastic integrals of random step functions, where the integrands are functionals of the integrators. Several results are achieved concerning statistical inference of time inhomogeneous diffusion processes. There are new results on statistical questions concerning spatial prorecces both in stable and unstable cases, and both in discrete and continuous time. We derived exact formulas for the Fourier transform of Gaussian measures on the Heisenberg group. We obtained new central limit theorems for locally compact Abelian groups

The handaxe and the microscope: individual and social learning in a multidimensional model of adaptation

When individuals learn by trial-and-error, they perform randomly chosen actions and then reinforce those actions that led to a high payoff. However, individuals do not always have to physically perform an action in order to evaluate its consequences. Rather, they may be able to mentally simulate actions and their consequences without actually performing them. Such fictitious learners can select actions with high payoffs without making long chains of trial-and-error learning. Here, we analyze the evolution of an n-dimensional cultural trait (or artifact) by learning, in a payoff landscape with a single optimum. We derive the stochastic learning dynamics of the distance to the optimum in trait space when choice between alternative artifacts follows the standard logit choice rule. We show that for both trial-and-error and fictitious learners, the learning dynamics stabilize at an approximate distance of root n/(2 lambda(e)) away from the optimum, where lambda(e) is an effective learning performance parameter depending on the learning rule under scrutiny. Individual learners are thus unlikely to reach the optimum when traits are complex (n large), and so face a barrier to further improvement of the artifact. We show, however, that this barrier can be significantly reduced in a large population of learners performing payoff-biased social learning, in which case lambda(e) becomes proportional to population size. Overall, our results illustrate the effects of errors in learning, levels of cognition, and population size for the evolution of complex cultural traits. (C) 2013 Elsevier Inc. All rights reserved

Markovian bridges: Weak continuity and pathwise constructions

A Markovian bridge is a probability measure taken from a disintegration of the law of an initial part of the path of a Markov process given its terminal value. As such, Markovian bridges admit a natural parameterization in terms of the state space of the process. In the context of Feller processes with continuous transition densities, we construct by weak convergence considerations the only versions of Markovian bridges which are weakly continuous with respect to their parameter. We use this weakly continuous construction to provide an extension of the strong Markov property in which the flow of time is reversed. In the context of self-similar Feller process, the last result is shown to be useful in the construction of Markovian bridges out of the trajectories of the original process.Comment: Published in at http://dx.doi.org/10.1214/10-AOP562 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sample Path Analysis of Integrate-and-Fire Neurons

Computational neuroscience is concerned with answering two intertwined questions that are based on the assumption that spatio-temporal patterns of spikes form the universal language of the nervous system. First, what function does a specific neural circuitry perform in the elaboration of a behavior? Second, how do neural circuits process behaviorally-relevant information? Non-linear system analysis has proven instrumental in understanding the coding strategies of early neural processing in various sensory modalities. Yet, at higher levels of integration, it fails to help in deciphering the response of assemblies of neurons to complex naturalistic stimuli. If neural activity can be assumed to be primarily driven by the stimulus at early stages of processing, the intrinsic activity of neural circuits interacts with their high-dimensional input to transform it in a stochastic non-linear fashion at the cortical level. As a consequence, any attempt to fully understand the brain through a system analysis approach becomes illusory. However, it is increasingly advocated that neural noise plays a constructive role in neural processing, facilitating information transmission. This prompts to gain insight into the neural code by studying the stochasticity of neuronal activity, which is viewed as biologically relevant. Such an endeavor requires the design of guiding theoretical principles to assess the potential benefits of neural noise. In this context, meeting the requirements of biological relevance and computational tractability, while providing a stochastic description of neural activity, prescribes the adoption of the integrate-and-fire model. In this thesis, founding ourselves on the path-wise description of neuronal activity, we propose to further the stochastic analysis of the integrate-and fire model through a combination of numerical and theoretical techniques. To begin, we expand upon the path-wise construction of linear diffusions, which offers a natural setting to describe leaky integrate-and-fire neurons, as inhomogeneous Markov chains. Based on the theoretical analysis of the first-passage problem, we then explore the interplay between the internal neuronal noise and the statistics of injected perturbations at the single unit level, and examine its implications on the neural coding. At the population level, we also develop an exact event-driven implementation of a Markov network of perfect integrate-and-fire neurons with both time delayed instantaneous interactions and arbitrary topology. We hope our approach will provide new paradigms to understand how sensory inputs perturb neural intrinsic activity and accomplish the goal of developing a new technique for identifying relevant patterns of population activity. From a perturbative perspective, our study shows how injecting frozen noise in different flavors can help characterize internal neuronal noise, which is presumably functionally relevant to information processing. From a simulation perspective, our event-driven framework is amenable to scrutinize the stochastic behavior of simple recurrent motifs as well as temporal dynamics of large scale networks under spike-timing-dependent plasticity