Diese Arbeit entwickelt eine Methode, Funktionalgleichungen der Art g(g(x))=f(x) bzw. g^n(x)=f(x) mit Hilfe neuronaler Netze zu lösen. Gesucht ist eine Funktion g(x), die mehrfach hintereinandergeschaltet genau einer gegebenen Funktion f(x) entspricht. Man nennt g=f^1/n eine iterative Wurzel oder fraktionale Iteration von f. Lösungen für g zu finden, stellt das inverse Problem der Iteration dar oder die Erweiterung der Wurzel- bzw. Potenzoperation auf die Funktionsalgebra. Geschlossene Ausdrücke für Funktionswurzeln einer gegebenen Funktion zu finden, ist in der Regel nicht möglich oder sehr schwer. Numerische Verfahren sind nicht in allgemeiner Form beschrieben oder als Software vorhanden. Ausgehend von der Fähigkeit eines neuronalen Netzes, speziell des mehrschichtigen Perzeptrons, durch Training eine gegebene Funktion f(x) zu approximieren, erlaubt eine spezielle Topologie des Netzes auch die Berechnung von fraktionalen Iterationen von f. Ein solches Netz besteht aus n identischen, hintereinandergeschalteten Teilnetzen, die, wenn das Gesamtnetz f approximiert, jedes für sich g = f^1/n annähern. Es ist lediglich beim Training des Netzes darauf zu achten, dass die korrespondierenden Gewichte aller Teilnetze den gleichen Wert annehmen. Dazu werden mehrere Verfahren entwickelt: Lernen nur im letzten Teilnetz und Kopieren der Gewichte auf die anderen Teile, Angleichen der Teilnetze durch Kopplungsfaktoren oder Einführung eines Fehlerterms, der Unterschiede in den Teilnetzen bestraft. Als weitere Näherungslösung wird ein iteriertes lineares Modell entwickelt, das durch ein herkömmliches neuronales Netz mit hoher Approximationsgüte für nichtlineare Zusammenhänge korrigiert wird. Als Anwendung ist konkret die Modellierung der Bandprofilentwicklung beim Warmwalzen von Stahlblech gegeben. Einige Zentimeter dicke Stahlblöcke werden in einer Walzstraße von mehreren gleichartigen, hintereinanderliegenden Walzgerüsten zu Blechen von wenigen Millimetern Dicke gewalzt. Neben der Dicke ist das Profil - der Dickenunterschied zwischen Bandmitte und Rand - eine wichtige Qualitätsgröße. Sie kann vor und hinter der Fertigstraße gemessen werden, aus technischen Gründen aber nicht zwischen den Walzgerüsten. Eine genaue Kenntnis ist jedoch aus produktionstechnischen Gründen wichtig. Der Stand der Technik ist die Berechnung dieser Zwischenprofile durch das wiederholte Durchrechnen eines mathematischen Modells des Walzvorganges für jedes Gerüst und eine ständige Anpassung von adaptiven Termen dieses Modells an die Messdaten. Es wurde gezeigt, dass mit einem adaptiven neuronalen Netz, das mit Eingangs- und Ausgangsprofil sowie allen vorhandenen Kenn- und Stellgrößen trainiert wird, die Vorausberechnung des Endprofils mit deutlich höherer Genauigkeit vorgenommen werden kann. Das Problem ist, dass dieses Netz die Übertragungsfunktion der gesamten Straße repräsentiert, Zwischenprofile können nicht ausgegeben werden. Daher wird der Versuch gemacht, beide Eigenschaften zu verbinden: Die genaue Endprofilmodellierung eines neuronalen Netzes wird kombiniert mit der Fähigkeit des iterierten Modells, Zwischenprofile zu berechnen. Dabei wird der in Form von Messdaten bekannte gesamte Prozess als iterierte Verknüpfung von technisch identischen Teilprozessen angesehen. Die Gewinnung eines Modells des Einzelprozesses entspricht damit der Berechnung der iterativen Wurzel des Gesamtprozesses.

In the first part of this thesis, we generalize a coding theorem and a converse of Kadota and Wyner (1972) to abstract channels with time structure. As a main contribution we prove the coding theorem for a significantly weaker condition on the channel output memory, called total ergodicity for block-i.i.d. inputs. We achieve this result mainly by introducing an alternative characterization of information rate capacity. We show that the ψ-mixing condition (asymptotic output-memorylessness), used by Kadota and Wyner, is quite restrictive, in particular for the important class of Gaussian channels. In fact, we prove that for Gaussian channels the ψ-mixing condition is equivalent to finite output memory. Moreover, we derive a weak converse for all stationary channels with time structure. Intersymbol interference as well as input constraints are taken into account in a flexible way. Due to the direct use of outer measures and a derivation of an adequate version of Feinstein’s lemma we are able to avoid the standard extension of the channel input σ-algebra and obtain a more transparent derivation. We aim at a presentation from an operational perspective and consider an abstract framework, which enables us to treat discrete- and continuous-time channels in a unified way. In the second part, we systematically analyze infinite output memory conditions for abstract channels with time structure. We exploit the connections to the rich field of strongly mixing random processes to derive a hierarchy for the nonequivalent infinite channel output memory conditions in terms of a sequence of implications. The ergodic-theoretic memory condition used in the proof of the coding theorem and the ψ-mixing condition employed by Kadota and Wyner (1972) are shown to be part of this taxonomy. In addition, we specify conditions for the channel under which memory properties of a random process are invariant when the process is passed through the channel. In the last part, we investigate cascade and integration channels with regard to mixing conditions as well as properties required in the context of the coding theorem. The results are useful to study many physically relevant channel models and allow a component-based analysis of the overall channel. We consider a number of examples including composed models and deterministic as well as random filter channels. Finally, an application of strong mixing conditions from statistical signal processing involving the Fourier transform of stationary random sequences is discussed and a list of further applications is given. / Im ersten Teil der Arbeit wird ein Kodierungstheorem und ein dazugehöriges Umkehrtheorem von Kadota und Wyner (1972) für abstrakte Kanäle mit Zeitstruktur verallgemeinert. Als wesentlichster Beitrag wird das Kodierungstheorem für eine signifikant schwächere Bedingung an das Kanalausgangsgedächtnis bewiesen, die sogenannte totale Ergodizität für block-i.i.d. Eingaben. Dieses Ergebnis wird hauptsächlich durch eine alternative Charakterisierung der Informationsratenkapazität erreicht. Es wird gezeigt, dass die von Kadota und Wyner verwendete ψ-Mischungsbedingung (asymptotische Gedächtnislosigkeit am Kanalausgang) recht einschränkend ist, insbesondere für die wichtige Klasse der Gaußkanäle. In der Tat, für Gaußkanäle wird bewiesen, dass die ψ-Mischungsbedingung äquivalent zu endlichem Gedächtnis am Kanalausgang ist. Darüber hinaus wird eine schwache Umkehrung für alle stationären Kanäle mit Zeitstruktur bewiesen. Sowohl Intersymbolinterferenz als auch Eingabebeschränkungen werden in allgemeiner und flexibler Form berücksichtigt. Aufgrund der direkten Verwendung von äußeren Maßen und der Herleitung einer angepassten Version von Feinsteins Lemma ist es möglich, auf die Standarderweiterung der σ-Algebra am Kanaleingang zu verzichten, wodurch die Darstellungen transparenter und einfacher werden. Angestrebt wird eine operationelle Perspektive. Die Verwendung eines abstrakten Modells erlaubt dabei die einheitliche Betrachtung von zeitdiskreten und zeitstetigen Kanälen. Für abstrakte Kanäle mit Zeitstruktur werden im zweiten Teil der Arbeit Bedingungen für ein unendliches Gedächtnis am Kanalausgang systematisch analysiert. Unter Ausnutzung der Zusammenhänge zu dem umfassenden Gebiet der stark mischenden zufälligen Prozesse wird eine Hierarchie in Form einer Folge von Implikationen zwischen den verschiedenen Gedächtnisvarianten hergeleitet. Die im Beweis des Kodierungstheorems verwendete ergodentheoretische Gedächtniseigenschaft und die ψ-Mischungsbedingung von Kadota und Wyner (1972) sind dabei Bestandteil der hergeleiteten Systematik. Weiterhin werden Bedingungen für den Kanal spezifiziert, unter denen Eigenschaften von zufälligen Prozessen am Kanaleingang bei einer Transformation durch den Kanal erhalten bleiben. Im letzten Teil der Arbeit werden sowohl Integrationskanäle als auch Hintereinanderschaltungen von Kanälen in Bezug auf Mischungsbedingungen sowie weitere für das Kodierungstheorem relevante Kanaleigenschaften analysiert. Die erzielten Ergebnisse sind nützlich bei der Untersuchung vieler physikalisch relevanter Kanalmodelle und erlauben eine komponentenbasierte Betrachtung zusammengesetzter Kanäle. Es wird eine Reihe von Beispielen untersucht, einschließlich deterministischer Kanäle, zufälliger Filter und daraus zusammengesetzter Modelle. Abschließend werden Anwendungen aus weiteren Gebieten, beispielsweise der statistischen Signalverarbeitung, diskutiert. Insbesondere die Fourier-Transformation stationärer zufälliger Prozesse wird im Zusammenhang mit starken Mischungsbedingungen betrachtet.

The first part analyzes an Endogenous Business Cycle model with embodied technological change. Households take an optimal decision about their spending for consumption and financing of R&D. The probability of a technology invention occurring is an increasing function of aggregate R&D expenditure in the whole economy. New technologies bring higher productivity, but rather than applying to the whole capital stock, they require a new vintage of capital, which first has to be accumulated before the productivity gain can be realized. The model offers some valuable features: Firstly, the response of output following a technology shock is very gradual; there are no jumps. Secondly, R&D is an ongoing activity; there are no distinct phases of research and production. Thirdly, R&D expenditure is pro-cyclical and the real interest rate is counter-cyclical. Finally, long-run growth is without scale effects. The second part analyzes a RBC model in continuous time featuring deterministic incremental development of technology and stochastic fundamental inventions arriving according to a Poisson process. In a special case an analytical solution is presented. In the general case a delay differential equation (DDE) has to be solved. Standard numerical solution methods fail, because the steady state is path dependent. A new solution method is presented which is based on a modified method of steps for DDEs. It provides not only approximations but also upper and lower bounds for optimal consumption path and steady state. Furthermore, analytical expressions for the long-term equilibrium distributions of the stationary variables of the model are presented. The distributions can be described as extended Beta distributions. This is deduced from a methodical result about a delay extension of the Pearson system.

This thesis investigates the extension of the well-known canonical correlation analysis for random elements on abstract real measurable Hilbert spaces. One focus is on the application of this extension to the calculation of information-theoretical quantities on finite time intervals. Analytical approaches for the calculation of the mutual information and the information density between Gaussian distributed random elements on arbitrary real measurable Hilbert spaces are derived. With respect to mutual information, the results obtained are comparable to [4] and [1] (Baker, 1970, 1978). They can also be seen as a generalization of earlier findings in [20] (Gelfand and Yaglom, 1958). In addition, some of the derived equations for calculating the information density, its characteristic function and its n-th central moments extend results from [45] and [44] (Pinsker, 1963, 1964). Furthermore, explicit examples for the calculation of the mutual information, the characteristic function of the information density as well as the n-th central moments of the information density for the important special case of an additive Gaussian channel with Gaussian distributed input signal with rational spectral density are elaborated, on the one hand for white Gaussian noise and on the other hand for Gaussian noise with rational spectral density. These results extend the corresponding concrete examples for the calculation of the mutual information from [20] (Gelfand and Yaglom, 1958) as well as [28] and [29] (Huang and Johnson, 1963, 1962).:Kurzfassung Abstract Notations Abbreviations 1 Introduction 1.1 Software Used 2 Mathematical Background 2.1 Basic Notions of Measure and Probability Theory 2.1.1 Characteristic Functions 2.2 Stochastic Processes 2.2.1 The Consistency Theorem of Daniell and Kolmogorov 2.2.2 Second Order Random Processes 2.3 Some Properties of Fourier Transforms 2.4 Some Basic Inequalities 2.5 Some Fundamentals in Functional Analysis 2.5.1 Hilbert Spaces 2.5.2 Linear Operators on Hilbert Spaces 2.5.3 The Fréchet-Riesz Representation Theorem 2.5.4 Adjoint and Compact Operators 2.5.5 The Spectral Theorem for Compact Operators 3 Mutual Information and Information Density 3.1 Mutual Information 3.2 Information Density 4 Probability Measures on Hilbert Spaces 4.1 Measurable Hilbert Spaces 4.2 The Characteristic Functional 4.3 Mean Value and Covariance Operator 4.4 Gaussian Probability Measures on Hilbert Spaces 4.5 The Product of Two Measurable Hilbert Spaces 4.5.1 The Product Measure 4.5.2 Cross-Covariance Operator 5 Canonical Correlation Analysis on Hilbert Spaces 5.1 The Hellinger Distance and the Theorem of Kakutani 5.2 Canonical Correlation Analysis on Hilbert Spaces 5.3 The Theorem of Hájek and Feldman 6 Mutual Information and Information Density Between Gaussian Measures 6.1 A General Formula for Mutual Information and Information Density for Gaussian Random Elements 6.2 Hadamard’s Factorization Theorem 6.3 Closed Form Expressions for Mutual Information and Related Quantities 6.4 The Discrete-Time Case 6.5 The Continuous-Time Case 6.6 Approximation Error 7 Additive Gaussian Channels 7.1 Abstract Channel Model and General Definitions 7.2 Explicit Expressions for Mutual Information and Related Quantities 7.2.1 Gaussian Random Elements as Input to an Additive Gaussian Channel 8 Continuous-Time Gaussian Channels 8.1 White Gaussian Channels 8.1.1 Two Simple Examples 8.1.2 Gaussian Input with Rational Spectral Density 8.1.3 A Method of Youla, Kadota and Slepian 8.2 Noise and Input Signal with Rational Spectral Density 8.2.1 Again a Method by Slepian and Kadota Bibliography / Diese Arbeit untersucht die Erweiterung der bekannten kanonischen Korrelationsanalyse (canonical correlation analysis) für Zufallselemente auf abstrakten reellen messbaren Hilberträumen. Ein Schwerpunkt liegt dabei auf der Anwendung dieser Erweiterung zur Berechnung informationstheoretischer Größen auf endlichen Zeitintervallen. Analytische Ansätze für die Berechnung der Transinformation und der Informationsdichte zwischen gaußverteilten Zufallselementen auf beliebigen reelen messbaren Hilberträumen werden hergeleitet. Bezüglich der Transinformation sind die gewonnenen Resultate vergleichbar zu [4] und [1] (Baker, 1970, 1978). Sie können auch als Verallgemeinerung früherer Erkenntnisse aus [20] (Gelfand und Yaglom, 1958) aufgefasst werden. Zusätzlich erweitern einige der hergeleiteten Formeln zur Berechnung der Informationsdichte, ihrer charakteristischen Funktion und ihrer n-ten zentralen Momente Ergebnisse aus [45] und [44] (Pinsker, 1963, 1964). Weiterhin werden explizite Beispiele für die Berechnung der Transinformation, der charakteristischen Funktion der Informationsdichte sowie der n-ten zentralen Momente der Informationsdichte für den wichtigen Spezialfall eines additiven Gaußkanals mit gaußverteiltem Eingangssignal mit rationaler Spektraldichte erarbeitet, einerseits für gaußsches weißes Rauschen und andererseits für gaußsches Rauschen mit einer rationalen Spektraldichte. Diese Ergebnisse erweitern die entsprechenden konkreten Beispiele zur Berechnung der Transinformation aus [20] (Gelfand und Yaglom, 1958) sowie [28] und [29] (Huang und Johnson, 1963, 1962).:Kurzfassung Abstract Notations Abbreviations 1 Introduction 1.1 Software Used 2 Mathematical Background 2.1 Basic Notions of Measure and Probability Theory 2.1.1 Characteristic Functions 2.2 Stochastic Processes 2.2.1 The Consistency Theorem of Daniell and Kolmogorov 2.2.2 Second Order Random Processes 2.3 Some Properties of Fourier Transforms 2.4 Some Basic Inequalities 2.5 Some Fundamentals in Functional Analysis 2.5.1 Hilbert Spaces 2.5.2 Linear Operators on Hilbert Spaces 2.5.3 The Fréchet-Riesz Representation Theorem 2.5.4 Adjoint and Compact Operators 2.5.5 The Spectral Theorem for Compact Operators 3 Mutual Information and Information Density 3.1 Mutual Information 3.2 Information Density 4 Probability Measures on Hilbert Spaces 4.1 Measurable Hilbert Spaces 4.2 The Characteristic Functional 4.3 Mean Value and Covariance Operator 4.4 Gaussian Probability Measures on Hilbert Spaces 4.5 The Product of Two Measurable Hilbert Spaces 4.5.1 The Product Measure 4.5.2 Cross-Covariance Operator 5 Canonical Correlation Analysis on Hilbert Spaces 5.1 The Hellinger Distance and the Theorem of Kakutani 5.2 Canonical Correlation Analysis on Hilbert Spaces 5.3 The Theorem of Hájek and Feldman 6 Mutual Information and Information Density Between Gaussian Measures 6.1 A General Formula for Mutual Information and Information Density for Gaussian Random Elements 6.2 Hadamard’s Factorization Theorem 6.3 Closed Form Expressions for Mutual Information and Related Quantities 6.4 The Discrete-Time Case 6.5 The Continuous-Time Case 6.6 Approximation Error 7 Additive Gaussian Channels 7.1 Abstract Channel Model and General Definitions 7.2 Explicit Expressions for Mutual Information and Related Quantities 7.2.1 Gaussian Random Elements as Input to an Additive Gaussian Channel 8 Continuous-Time Gaussian Channels 8.1 White Gaussian Channels 8.1.1 Two Simple Examples 8.1.2 Gaussian Input with Rational Spectral Density 8.1.3 A Method of Youla, Kadota and Slepian 8.2 Noise and Input Signal with Rational Spectral Density 8.2.1 Again a Method by Slepian and Kadota Bibliography

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Given a full-range simply-invariant shift-invariant subspace <i>M</i> of the vector-valued <i>L²</i> space on the unit circle, the classical Beurling-Lax-Halmos (BLH) theorem obtains a unitary operator-valued function <i>W</i> so that <i>M</i> may be represented as the image of of the Hardy space <i>H²</i> on the disc under multiplication by <i>W</i>. The work of Ball-Helton later extended this result to find a single function representing a so-called dual shift-invariant pair of subspaces <i>(M,M^Ã)</i> which together form a direct-sum decomposition of <i>L²</i>. In the case where the pair <i>(M,M^Ã)</i> are finite-dimensional perturbations of the Hardy space <i>H²</i> and its orthogonal complement, Ball-Gohberg-Rodman obtained a transfer function realization for the representing function <i>W</i>; this realization was parameterized in terms of zero-pole data computed from the pair <i>(M,M^Ã)</i>. Later work by Ball-Raney extended this analysis to the case of nonrational functions <i>W</i> where the zero-pole data is taken in an infinite-dimensional operator theoretic sense. The current work obtains analogues of these various results for arbitrary dual shift-invariant pairs <i>(M,M^Ã)</i> of the <i>L²</i> spaces on the real line; here, shift-invariance refers to invariance under the translation group. These new results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems which have been codified in the book by Staffans. / Ph. D.

Anomale Diffusion ist ein weitverbreiteter Transportmechanismus, welcher für gewöhnlich mit ensemble-basierten Methoden experimentell untersucht wird. Motiviert durch den Fortschritt in der Einzelteilchenverfolgung, wo typischerweise Zeitmittelwerte bestimmt werden, entsteht die Frage nach der Ergodizität. Stimmen ensemble-gemittelte Größen und zeitgemittelte Größen überein, und wenn nicht, wie unterscheiden sie sich? In dieser Arbeit studieren wir verschiedene stochastische Modelle für anomale Diffusion bezüglich ihres ergodischen oder nicht-ergodischen Verhaltens hinsichtlich der mittleren quadratischen Verschiebung. Wir beginnen unsere Untersuchung mit integrierter Brownscher Bewegung, welche von großer Bedeutung für alle Systeme mit Impulsdiffusion ist. Für diesen Prozess stellen wir die ensemble-gemittelte quadratische Verschiebung und die zeitgemittelte quadratische Verschiebung gegenüber und charakterisieren insbesondere die Zufälligkeit letzterer. Im zweiten Teil bilden wir integrierte Brownsche Bewegung auf andere Modelle ab, um einen tieferen Einblick in den Ursprung des nicht-ergodischen Verhaltens zu bekommen. Dabei werden wir auf einen verallgemeinerten Lévy-Lauf geführt. Dieser offenbart interessante Phänomene, welche in der Literatur noch nicht beobachtet worden sind. Schließlich führen wir eine neue Größe für die Analyse anomaler Diffusionsprozesse ein, die Verteilung der verallgemeinerten Diffusivitäten, welche über die mittlere quadratische Verschiebung hinausgeht, und analysieren mit dieser ein oft verwendetes Modell der anomalen Diffusion, den subdiffusiven zeitkontinuierlichen Zufallslauf. / Anomalous diffusion is a widespread transport mechanism, which is usually experimentally investigated by ensemble-based methods. Motivated by the progress in single-particle tracking, where time averages are typically determined, the question of ergodicity arises. Do ensemble-averaged quantities and time-averaged quantities coincide, and if not, in what way do they differ? In this thesis, we study different stochastic models for anomalous diffusion with respect to their ergodic or nonergodic behavior concerning the mean-squared displacement. We start our study with integrated Brownian motion, which is of high importance for all systems showing momentum diffusion. For this process, we contrast the ensemble-averaged squared displacement with the time-averaged squared displacement and, in particular, characterize the randomness of the latter. In the second part, we map integrated Brownian motion to other models in order to get a deeper insight into the origin of the nonergodic behavior. In doing so, we are led to a generalized Lévy walk. The latter reveals interesting phenomena, which have never been observed in the literature before. Finally, we introduce a new tool for analyzing anomalous diffusion processes, the distribution of generalized diffusivities, which goes beyond the mean-squared displacement, and we analyze with this tool an often used model of anomalous diffusion, the subdiffusive continuous time random walk.

Um conjunto de modelos compostos de redes de filas em grades finitas servindo como ambientes aleatorios para um ou mais passeios aleatorios, que por sua vez podem afetar o comportamento das filas, e desenvolvido. Duas formas de interacao entre os passeios aleatorios sao consideradas. Para cada modelo, e provado que o processo Markoviano correspondente e recorrente positivo e reversivel. As equacoes de balanceamento detalhado sao analisadas para obter a forma funcional da medida invariante de cada modelo. Em todos os modelos analisados neste trabalho, a medida invariante em uma grade finita tem forma produto. Modelos de redes de filas como ambientes para multiplos passeios aleatorios sao estendidos a grades infinitas. Para cada modelo estendido, sao especificadas as condicoes para a existencia do processo estocastico na grade infinita. Alem disso, e provado que existe uma unica medida invariante na rede infinita cuja projecao em uma subgrade finita e dada pela medida correspondente de uma rede finita. Finalmente, e provado que essa medida invariante na rede infinita e reversivel. / A set of models composed of queueing networks serving as random environments for one or more random walks, which themselves can affect the behavior of the queues, is developed. Two forms of interaction between the random walkers are considered. For each model, it is proved that the corresponding Markov process is positive recurrent and reversible. The detailed balance equa- tions are analyzed to obtain the functional form of the invariant measure of each model. In all the models analyzed in the present work, the invariant measure on a finite lattice has product form. Models of queueing networks as environments for multiple random walks are extended to infinite lattices. For each model extended, the conditions for the existence of the stochastic process on the infinite lattice are specified. In addition, it is proved that there exists a unique invariant measure on the infinite network whose projection on a finite sublattice is given by the corresponding finite- network measure. Finally, it is proved that that invariant measure on the infinite lattice is reversible.

Um conjunto de modelos compostos de redes de filas em grades finitas servindo como ambientes aleatorios para um ou mais passeios aleatorios, que por sua vez podem afetar o comportamento das filas, e desenvolvido. Duas formas de interacao entre os passeios aleatorios sao consideradas. Para cada modelo, e provado que o processo Markoviano correspondente e recorrente positivo e reversivel. As equacoes de balanceamento detalhado sao analisadas para obter a forma funcional da medida invariante de cada modelo. Em todos os modelos analisados neste trabalho, a medida invariante em uma grade finita tem forma produto. Modelos de redes de filas como ambientes para multiplos passeios aleatorios sao estendidos a grades infinitas. Para cada modelo estendido, sao especificadas as condicoes para a existencia do processo estocastico na grade infinita. Alem disso, e provado que existe uma unica medida invariante na rede infinita cuja projecao em uma subgrade finita e dada pela medida correspondente de uma rede finita. Finalmente, e provado que essa medida invariante na rede infinita e reversivel. / A set of models composed of queueing networks serving as random environments for one or more random walks, which themselves can affect the behavior of the queues, is developed. Two forms of interaction between the random walkers are considered. For each model, it is proved that the corresponding Markov process is positive recurrent and reversible. The detailed balance equa- tions are analyzed to obtain the functional form of the invariant measure of each model. In all the models analyzed in the present work, the invariant measure on a finite lattice has product form. Models of queueing networks as environments for multiple random walks are extended to infinite lattices. For each model extended, the conditions for the existence of the stochastic process on the infinite lattice are specified. In addition, it is proved that there exists a unique invariant measure on the infinite network whose projection on a finite sublattice is given by the corresponding finite- network measure. Finally, it is proved that that invariant measure on the infinite lattice is reversible.

Subprime residential mortgage loan securitization and its associated risks have been a major topic of discussion since the onset of the subprime mortgage crisis (SMC) in 2007. In this regard, the thesis addresses the issues of subprime residential mortgage loan (RML) securitization in discrete-, continuous-and discontinuous-time and their connections with the SMC. In this regard, the main issues to be addressed are discussed in Chapters 2, 3 and 4. In Chapter 2, we investigate the risk allocation choices of an investing bank (IB) that has to decide between risky securitized subprime RMLs and riskless Treasuries. This issue is discussed in a discrete-time framework with IB being considered to be regret- and risk-averse before and during the SMC, respectively. We conclude that if IB takes regret into account it will be exposed to higher risk when the difference between the expected returns on securitized subprime RMLs and Treasuries is small. However, there is low risk exposure when this difference is high. Furthermore, we assess how regret can influence IB's view - as a swap protection buyer - of the rate of return on credit default swaps (CDSs), as measured by the premium based on default swap spreads. We find that before the SMC, regret increases IB's willingness to pay lower premiums for CDSs when its securitized RML portfolio is considered to be safe. On the other hand, both risk- and regret-averse IBs pay the same CDS premium when their securitized RML portfolio is considered to be risky. Chapter 3 solves a stochastic optimal credit default insurance problem in continuous-time that has the cash outflow rate for satisfying depositor obligations, the investment in securitized loans and credit default insurance as controls. As far as the latter is concerned, we compute the credit default swap premium and accrued premium by considering the credit rating of the securitized mortgage loans. In Chapter 4, we consider a problem of IB investment in subprime residential mortgage-backed securities (RMBSs) and Treasuries in discontinuous-time. In order to accomplish this, we develop a Levy process-based model of jump diffusion-type for IB's investment in subprime RMBSs and Treasuries. This model incorporates subprime RMBS losses which can be associated with credit risk. Furthermore, we use variance to measure such risk, and assume that the risk is bounded by a certain constraint. We are now able to set-up a mean-variance optimization problem for IB's investment which determines the optimal proportion of funds that needs to be invested in subprime RMBSs and Treasuries subject to credit risk measured by the variance of IE's investment. In the sequel, we also consider a mean swaps-at-risk (SaR) optimization problem for IB's investment which determines the optimal portfolio which consists of subprime RMBSs and Treasuries subject to the protection by CDSs required against the possible losses. In this regard, we define SaR as indicative to IB on how much protection from swap protection seller it must have in order to cover the losses that might occur from credit events. Moreover, SaR is expressed in terms of Value-at-Risk (VaR). Finally, Chapter 5 provides an analysis of discrete-, continuous- and discontinuous-time models for subprime RML securitization discussed in the aforementioned chapters and their connections with the SMC. The work presented in this thesis is based on 7 peer-reviewed international journal articles (see [25], [44], [45], [46], [47], [48] and [55]), 4 peer-reviewed chapters in books (see [42], [50j, [51J and [52]) and 2 peer-reviewed conference proceedings papers (see [11] and [12]). Moreover, the article [49] is currently being prepared for submission to an lSI accredited journal. / Thesis (Ph.D. (Applied Mathematics))--North-West University, Potchefstroom Campus, 2010.