Financial cycles, credit networks and macroeconomic fluctuations: multi-scale stochastic models and wavelet analysis

This project focuses on the macroeconomics of financial cycles. Usually defined in terms of self-reinforcing interactions between perceptions of value and risk, attitudes towards risk and financing constraints, which translate into booms followed by bust, the recent empirical literature has recurred to two approaches \u2013 turning point analysis and frequency-based filters - applied to measures of credit and asset prices to pose a number of stylized facts. First, financial cycles tend to display a greater amplitude and a lower frequency in comparison to business cycles, with peaks associated with systemic crises. Second, financial cycles depend on policy regimes and on the pace of financial innovations, leading to a wide cross-country heterogeneity and a time-varying degree of global synchronization. The latter point is clearly related to the structural transformations occurred in financial systems over the last three decades, like the cumulative integration of traditional banking with capital market developments and the increasing degree of interconnections among financial institutions. However, to date very little is known about determinants and mechanisms behind financial cycles, and on how they interact with business cycles and medium-to-long-run macroeconomic performance. In this project we plan to research along three dimensions: i) measurement issues, in order to provide a comprehensive assessment of the evolution of co-movements between financial and real variables across a sample of financial developed countries, both over time and at different frequencies; ii) theoretical issues, aimed at exploring under what circumstances the network of interconnections among financial intermediaries and between intermediaries and non-financial borrowers might evolve cyclically, contributing this way to regulate the incentives agents have in taking risks, and to set the importance of credit and financial frictions in accounting for time-varying misallocations of resources; iii) policy issues, given the role assigned by international supervisory bodies to a proper characterization and knowledge of the financial cycle as a prerequisite for the macro-prudential regulation of banks, and the scope of monetary policy in promoting financial stability in addition to the typical mandate of price stability. Our task requires the employment of a new approach to macroeconomic analysis, diverse analytical tools and one unifying economic principle. As regards the latter, our focal point is the notion of risk externalities, across financial institutions and between the financial sector and the real economy. The set of tools we plan to employ spans from wavelets methods to multi-scale models in continuous time, and from strategic network formation to agent-based computational techniques. All these tools are instrumental in building and estimating macroeconomic models characterized by interrelated markets operating at different time scales

Levenberg-Marquardt Algorithms for Nonlinear Equations, Multi-objective Optimization, and Complementarity Problems

The Levenberg-Marquardt algorithm is a classical method for solving nonlinear systems of equations that can come from various applications in engineering and economics. Recently, Levenberg-Marquardt methods turned out to be a valuable principle for obtaining fast convergence to a solution of the nonlinear system if the classical nonsingularity assumption is replaced by a weaker error bound condition. In this way also problems with nonisolated solutions can be treated successfully. Such problems increasingly arise in engineering applications and in mathematical programming. In this thesis we use Levenberg-Marquardt algorithms to deal with nonlinear equations, multi-objective optimization and complementarity problems. We develop new algorithms for solving these problems and investigate their convergence properties. For sufficiently smooth nonlinear equations we provide convergence results for inexact Levenberg-Marquardt type algorithms. In particular, a sharp bound on the maximal level of inexactness that is sufficient for a quadratic (or a superlinear) rate of convergence is derived. Moreover, the theory developed is used to show quadratic convergence of a robust projected Levenberg-Marquardt algorithm. The use of Levenberg-Marquardt type algorithms for unconstrained multi-objective optimization problems is investigated in detail. In particular, two globally and locally quadratically convergent algorithms for these problems are developed. Moreover, assumptions under which the error bound condition for a Pareto-critical system is fulfilled are derived. We also treat nonsmooth equations arising from reformulating complementarity problems by means of NCP functions. For these reformulations, we show that existing smoothness conditions are not satisfied at degenerate solutions. Moreover, we derive new results for positively homogeneous functions. The latter results are used to show that appropriate weaker smoothness conditions (enabling a local Q-quadratic rate of convergence) hold for certain reformulations.Der Levenberg-Marquardt-Algorithmus ist ein klassisches Verfahren zur Lösung von nichtlinearen Gleichungssystemen, welches in verschiedenen Anwendungen der Ingenieur-und Wirtschaftswissenschaften vorkommen kann. Kürzlich, erwies sich das Verfahren als ein wertvolles Instrument für die Gewährleistung einer schnelleren Konvergenz für eine Lösung des nichtlinearen Systems, wenn die klassische nichtsinguläre Annahme durch eine schwächere Fehlerschranke der eingebundenen Bedingung ersetzt wird. Auf diese Weise, lassen sich ebenfalls Probleme mit nicht isolierten Lösungen erfolgreich behandeln. Solche Probleme ergeben sich zunehmend in den praktischen, ingenieurwissenschaftlichen Anwendungen und in der mathematischen Programmierung. In dieser Arbeit verwenden wir Levenberg-Marquardt- Algorithmus für nichtlinearere Gleichungen, multikriterielle Optimierung - und nichtlineare Komplementaritätsprobleme. Wir entwickeln neue Algorithmen zur Lösung dieser Probleme und untersuchen ihre Konvergenzeigenschaften. Für ausreichend differenzierbare nichtlineare Gleichungen, analysieren und bieten wir Konvergenzergebnisse für ungenaue Levenberg-Marquardt-Algorithmen Typen. Insbesondere, bieten wir eine strenge Schranke für die maximale Höhe der Ungenauigkeit, die ausreichend ist für eine quadratische (oder eine superlineare) Rate der Konvergenz. Darüber hinaus, die entwickelte Theorie wird verwendet, um quadratische Konvergenz eines robusten projizierten Levenberg-Marquardt-Algorithmus zu zeigen. Die Verwendung von Levenberg-Marquardt-Algorithmen Typen für unbeschränkte multikriterielle Optimierungsprobleme im Detail zu untersucht. Insbesondere sind zwei globale und lokale quadratische konvergente Algorithmen für multikriterielle Optimierungsprobleme entwickelt worden. Die Annahmen wurden hergeleitet, unter welche die Fehlerschranke der eingebundenen Bedingung für ein Pareto-kritisches System erfüllt ist. Wir behandeln auch nicht differenzierbare nichtlineare Gleichungen aus Umformulierung der nichtlinearen Komplementaritätsprobleme durch NCP-Funktionen. Wir zeigen für diese Umformulierungen, dass die bestehenden differenzierbaren Bedingungen nicht zufrieden mit degenerierten Lösungen sind. Außerdem, leiten wir neue Ergebnisse für positiv homogene NCP-Funktionen. Letztere Ergebnisse werden verwendet um zu zeigen, dass geeignete schwächeren differenzierbare Bedingungen (so dass eine lokale Q-quadratische Konvergenzgeschwindigkeit ermöglichen) für bestimmte Umformulierungen gelten