Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Extending Implicit Computational Complexity and Abstract Machines to Languages with Control

The Curry-Howard isomorphism is the idea that proofs in natural deduction can be put in correspondence with lambda terms in such a way that this correspondence is preserved by normalization. The concept can be extended from Intuitionistic Logic to other systems, such as Linear Logic. One of the nice conseguences of this isomorphism is that we can reason about functional programs with formal tools which are typical of proof systems: such analysis can also include quantitative qualities of programs, such as the number of steps it takes to terminate. Another is the possiblity to describe the execution of these programs in terms of abstract machines. In 1990 Griffin proved that the correspondence can be extended to Classical Logic and control operators. That is, Classical Logic adds the possiblity to manipulate continuations. In this thesis we see how the things we described above work in this larger context.

How to cope with air transport disruptions: airport airside resilience and vulnerability

The efficiency of airport airside operations is often compromised by unplanned disruptive events of different kinds, such as bad weather, strikes or technical failures, which negatively influence the punctuality and regularity of operations, causing serious delays and unexpected congestion. They may provoke important impacts and economic losses on passengers, airlines and airport operators, and consequences may propagate in the air network throughout different airports. In order to identify strategies to cope with such events and minimize their impacts, it is crucial to understand how disruptive events affect airports’ performance. The research field related with the risk of severe air transport network disruptions and their impact on society is related to the concepts of vulnerability and resilience. The main objective of this project is to provide a framework that allows to evaluate performance losses and consequences due to unexpected disruptions affecting airport airside operations, supporting the development of a methodology for estimating vulnerability and resilience indicators for airport airside operations. The methodology proposed comprises three phases. In the first phase, airside operations are modelled in both the baseline and disrupted scenarios. The model includes all main airside processes and takes into consideration the uncertainties and dynamics of the system. In the second phase, the model is implemented by using a generic simulation software, AnyLogic. Vulnerability is evaluated by taking into consideration the costs related to flight delays, cancellations and diversions; resilience is determined as a function of the loss of capacity during the entire period of disruption. In the third phase, a Bayesian Network is built in which uncertain variables refer to airport characteristics and disruption type. The Bayesian Network expresses the conditional dependence among these variables and allows to predict the impacts of disruptions on an airside system, determining the elements which influence the system resilience the most.

The Financial Services Regime in International Economic Law: on the Difficult Relationship between Trade Liberalization and Prudential Measures

The PhD thesis analyses the financial services regime in international economic law from the perspective of the difficult relationship between trade liberalisation and prudential measures. Financial stability plays a fundamental role for the well-being and well-functioning of the global economy, but, it is at the same time a complex sector to regulate and supervise and, especially after the 2007-08 economic crisis, States have tightened up their regulation of financial services, introducing more severe and protectionist prudential measures. However, in an increasingly interconnected global economy, the harmonization of prudential regulation at the international level is an essential step to guarantee integrity, fairness and stability of financial markets and trade. The research analyses the tools at disposition to achieve this aim, the related problematic issues and the perspectives and possible solutions for the future, starting from the World Trade Organization (WTO) legal framework and its General Agreement on Trade in Services (GATS), devoted to discipline trade in services among the WTO Members. Then, the research moves to a second legal instrument, the Free Trade Agreements (FTAs), which has witnessed a remarkable spread in the last decades. Finally, the research addresses the international standards, developed by supranational entities and implemented by an increasing number of States, as they offer rules and guidelines adequate to update the international financial scenario. Nevertheless, the international standards alone cannot be the solution because, first, they are not mandatory, as governments decide voluntarily to apply them and, second, their decision-making process do not respect the requirements of transparency and representative membership. In light of this analysis, the thesis aims at providing an answer to its research question: how to give more certainty to States and economic operators in the planning of the domestic disciplines and business activities in order to provide a sound and stable international financial system.

Classical limit of the Nelson model

Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.

Legal Ontologies for Public Procurement Management

The thesis explores ways to formalize the legal knowledge concerning the public procurement domain by means of ontological patterns suitable, on one hand, to support awarding authorities in conducting procurement procedures and, on the other hand, to help citizens and economic operators in accessing procurement's notices and data. Such an investigation on the making up of conceptual models for the public procurement domain, in turn, inspires and motivates a reflection on the role of legal ontologies nowadays, as in the past, retracing the steps of the ``ontological legal thinking'' from Roman Law up to now. I try, at the same time, to forecast the impact, in terms of benefits, challenges and critical issues, of the application of computational models of Law in future e-Governance scenarios.

Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.