Model theory | EPFL Graph Search

In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit to cl

Coalgèbres d'Alexander-Whitney

Théophile Naïto

The goal of this work is to study Alexander-Whitney coalgebras (first defined in [HPST06]) from a topological point of view. An Alexander-Whitney coalgebra is a coassociative chain coalgebra over Z with an extra algebraic structure : the comultiplication must respect the coalgebra structure up to an infinite sequence of homotopies (this sequence is part of the data of the Alexander-Whitney coalgebra structure). Alexander-Whitney coalgebras are interesting for topologists because the normalized chain complex C(K) of a simplicial set K is endowed with an Alexander-Whitney coalgebra structure. This theorem is proved for the first time here (generalising a result proven in [HPST06]). This theorem gives the hope that the Alexander-Whitney coalgebra structure of C(K) contains interesting information that can be used to solve topological problems. This hope is strengthened by the success already obtained in the work of several topologists. Among others, [HPST06], [HL07], [Boy08], and [HR] use the Alexander-Whitney coalgebra structure of the normalized chains of a simplicial set in an essential way to solve topological problems. This thesis begins with some background material. In particular, the definition of a DCSH morphism between two coassociative chain coalgebras is recalled in complete detail. For example, signs are determined with great precision. Next we devote a chapter to the definition of Alexander-Whitney coalgebras and to their importance in topology. In the following chapter we begin the conceptual study of Alexander-Whitney coalgebras. A global study of these objects had not yet been carried out even if the Alexander-Whitney coalgebra structure has been studied and used in order to answer some specific questions. With the aim of studying Alexander-Whitney coalgebras in a nice setting, we develop an operadic description of these coalgebras in the following chapter. More precisely, we show that there is an explicit operad AW such that the coalgebras over this operad are exactly the Alexander-Whitney coalgebras. Furthermore, AW is shown to be a Hopf operad, so that the category formed by the Alexander-Whitney coalgebras is actually a monoidal category. These results are proven in a reasonably general framework. In fact, we associate an operad to each bimodule (over the associative operad) of a certain type, such that we get AW if this bimodule is well chosen. In particular, these results enable us to study Alexander-Whitney coalgebras from the standpoint of operads. This strategy is recognised to be successful in various mathematical situations, and especially in algebraic topology. Moreover, we develop a minimal model notion in the setting of right module over a chosen operad (which has to satisfy some reasonable conditions), with the aim of applying this result to the special case of the Alexander-Whitney coalgebras. This is possible because coalgebras over some fixed operad P can be seen as right modules over P. And the category of right modules over P has some nice features which do not appear to hold in the category of P-coalgebras. The inspiration for this part of our work comes from the notion of minimal model developed in the framework of rational homotopy theory. The two following facts show that it is reasonable to try to adapt some ideas of rational homotopy theory to the category of Alexander-Whitney coalgebras. A. There is a theorem that says that studying topological spaces up to rational equivalences is, essentially, equivalent to studying cocommutative chain coalgebras over the field of rational numbers. This is false if the ring of integers replaces the field of rational numbers, but Alexander-Whitney coalgebras are "almost" cocommutative in the sense which is explained in this thesis. B. It could be that the Alexander-Whitney coalgebra structure of the normalized chains of a simplicial set is weak enough to allow explicit computations. At least, it is clear that the Alexander-Whitney coalgebra structure on the normalized chains is far from being an E∞-structure (such a structure determines the homotopy type of the considered simplicial set, at least under some conditions). The chapter about minimal models in the framework of right modules over an operad includes an existence theorem and a discussion of the unicity of this model. In the second part of this chapter, we construct an explicit path-object in the model category of right modules over an operad. This path-object is then used to investigate the topologically relevant information that could stem from the minimal model in the case of the operad AW. Finally, we present and examine some interesting open questions about Alexander-Whitney coalgebras. These questions give a nice outlook on future research in this area.

EPFL2009

Bayesian risk analysis of financial time series

Christophe Osinski

This thesis is a contribution to financial statistics. One of the principal concerns of investors is the evaluation of portfolio risk. The notion of risk is vague, but in finance it is always linked to possible losses. In this thesis, we present some measures allowing the valuation of risk with the help of Bayesian methods. An exploratory analysis of data is presented to describe the sampling properties of financial time series. This analysis allows us to understand the origins of the daily returns studied in this thesis. Moreover, a discussion of different models is presented. These models make strong assumptions on investor behaviour, which are not always satisfied. This exploratory analysis shows some differences between the behaviour anticipated under equilibrium models, and that of real data. The Bayesian approach has been chosen because it allows one to incorporate all the variability, in particular that associated with model choice. The models studied in this thesis allow one to take heteroskedasticity into account, as well as particular shapes of the tails of returns. ARCH type models and models based on extreme value theory are studied. One original aspect of this thesis is its use of Bayesian analysis to detect change points in financial time series. We suppose that a market has two phases, and that it switches from a state to the other at random. Another new contribution is a model integrating heteroskedasticity and time dependence of extreme values, by superposition of the model proposed by Bortot and Coles (2003) and a GARCH process. This thesis uses simulation intensively for the estimation of risk measures. The drawback of simulation is the amount of time needed to obtain accurate estimates. However, simulation allows one to produce results when direct calculation is not feasible. For example, simulation allows one to compute risk estimates for time horizons greater than one day. The methods presented in this thesis are illustrated on simulated data, and on real data from European and American markets. This thesis involved the construction of a library containing C and S code to perform risk analysis using GARCH and extreme value theory models. The results show that model uncertainty can be incorporated, and that risk measures for time horizons greater than one can be obtained by simulation. The methods presented in this thesis have a natural representation involving conditioning. Thus, they permit the computation of both conditional and unconditional risk estimates. Three methods are described: the GARCH method; the two-state GARCH method; and the HBC method. Unconditional risk estimation using the GARCH method is satisfactory on data which seem stationary, but not reliable on data which are non-stationary, such as data with change points. The two-state GARCH model does a little better, but gives very satisfactory results when the risk is estimated conditionally on time. The HBC method does not give satisfactory results.

EPFL2006

Essays in Monetary Policy and Asset Pricing

Benoit Vincent Sylvain Cornet

This thesis uses machine learning techniques and text data to investigate the relationships that arise between the Fed and financial markets, and their consequences for asset prices.The first chapter, entitled Market Expectations and the Impact of Unconventional Monetary Policy: An Application to Twitter Data, is an answer to (Greenlaw et al., 2018), who show, by looking at a large set of monetary policy announcements made between November 2008 and December 2017 - FOMC meetings, release of minutes and speeches of the Fed Chair - that long term yields tended to increase following these events. By doing so, the authors challenge common wisdom according to which the central bank intervention in the aftermath of the financial crisis lowered long term rates (Gagnon, 2016). Using machine learning and twitter data, this chapter develops a novel measure of market expectations of monetary policy, and shows that the increase in yields was simply due to a marginal adjustment of market expectations following announcements being less dovish than expected.The second chapter, entitled Informational Feedback Loop, Monetary Policy Decisions and Asset Prices Dynamics, investigates the consequences of a Fed that uses (1) its own private signal and (2) fed funds futures to take its monetary policy decision. Fed funds futures aggre- gate private information received by financial markets participants - traders - but they also depend on traders' expectations about the Fed's behavior, which makes futures endogenous in the central bank decision. The theoretical model shows that the surprise generated by monetary policy announcements and the subsequent adjustment in short term U.S. treasury yields depend on the precision of the signals received by each agent. When the signal received by traders is more precise than the central bank's, the latter relies more on fed funds futures to take its decision, and the surprise and adjustment of short term yields are smaller. By contrast, long term yields adjust only because the announcement provides traders with new information about the state of the economy, by revealing the central bank's private signal. Finally, when the Fed is averse to financial markets volatility, it tends to put some weight on fed funds futures even if they are not informative about the state of the economy. The empirical part of the paper provides some evidence supporting these channels, by using a topic and tone approach (Hansen and McMahon, 2016) to extract the precision of the signals received by the central bank and traders from FOMC minutes and tweets respectively.

EPFL2022