Topological stability through tame retractions

A smooth map is said to be stable if small perturbations of the map only differ from the original one by a smooth change of coordinates. Smoothly stable maps are generic among the proper maps between given source and target manifolds when the source and target dimensions belong to the so-called nice dimensions, but outside this range of dimensions, smooth maps cannot generally be approximated by stable maps. This leads to the definition of topologically stable maps, where the smooth coordinate changes are replaced with homeomorphisms. The topologically stable maps are generic among proper maps for any dimensions of source and target. The purpose of this thesis is to investigate methods for proving topological stability by constructing extremely tame (E-tame) retractions onto the map in question from one of its smoothly stable unfoldings. In particular, we investigate how to use E-tame retractions from stable unfoldings to find topologically ministable unfoldings for certain weighted homogeneous maps or germs. Our first results are concerned with the construction of E-tame retractions and their relation to topological stability. We study how to construct the E-tame retractions from partial or local information, and these results form our toolbox for the main constructions. In the next chapter we study the group of right-left equivalences leaving a given multigerm f invariant, and show that when the multigerm is finitely determined, the group has a maximal compact subgroup and that the corresponding quotient is contractible. This means, essentially, that the group can be replaced with a compact Lie group of symmetries without much loss of information. We also show how to split the group into a product whose components only depend on the monogerm components of f. In the final chapter we investigate representatives of the E- and Z-series of singularities, discuss their instability and use our tools to construct E-tame retractions for some of them. The construction is based on describing the geometry of the set of points where the map is not smoothly stable, discovering that by using induction and our constructional tools, we already know how to construct local E-tame retractions along the set. The local solutions can then be glued together using our knowledge about the symmetry group of the local germs. We also discuss how to generalize our method to the whole E- and Z- series.

Essays on Banking Crises

This thesis aims at finding the role of deposit insurance scheme and central bank (CB) in keeping the banking system safe. The thesis also studies the factors associated with long-lasting banking crises. The first essay analyzes the effect of using explicit deposit insurance scheme (EDIS), instead of using implicit deposit insurance scheme (IDIS), on banking crises. The panel data for the period of 1980-2003 includes all countries for which the data on EDIS or IDIS exist. 70% of the countries in the sample are less developed countries (LDCs). About 55% of the countries adopting EDIS also come from LDCs. The major finding is that the using of EDIS increases the crisis probability at a strong significance level. This probability is greater if the EDIS is inefficiently designed allowing higher scope of moral hazard problem. Specifically, the probability is greater if the EDIS provides higher coverage to deposits and if it is less powerful from the legal point of view. This study also finds that the less developed a country is to handle EDIS, the higher the chance of banking crisis. Once the underdevelopment of an economy handling the EDIS is controlled, the EDIS separately is no longer a significant factor of banking crises. The second essay aims at determining whether a country s powerful CB can lessen the instability of the banking sector by minimizing the likelihood of a banking crisis. The data used include indicators of the CB s autonomy for a set of countries over the period of 1980-89. The study finds that in aggregate a more powerful CB lessens the probability of banking crisis. When the CB s authority is disentangled with respect to its responsibilities, the study finds that the longer tenure of CB s chief executive officer and the greater power of CB in assigning interest rate on government loans are necessary for reducing the probability of banking crisis. The study also finds that the probability of crisis reduces more if an autonomous CB can perform its duties in a country with stronger law and order tradition. The costs of long-lasting banking crises are high because both the depositors and the investors lose confidence in the banking system. For a rapid recovery of a crisis, the government very often undertakes one or more crisis resolution policy (CRP) measures. The third essay examines the CRP and other explanatory variables correlated with the durations of banking crises. The major finding is that the CRP measure allowing the regulation forbearance to keep the insolvent banks operative and the public debt relief program are respectively strongly and weakly significant to increase the durations of crises. Some other explanatory variables, which were found by previous studies to be related with the probability of crises to occur, are also correlated with the durations of crises.

Elements of Volatility at High Frequency

Volatility is central in options pricing and risk management. It reflects the uncertainty of investors and the inherent instability of the economy. Time series methods are among the most widely applied scientific methods to analyze and predict volatility. Very frequently sampled data contain much valuable information about the different elements of volatility and may ultimately reveal the reasons for time varying volatility. The use of such ultra-high-frequency data is common to all three essays of the dissertation. The dissertation belongs to the field of financial econometrics. The first essay uses wavelet methods to study the time-varying behavior of scaling laws and long-memory in the five-minute volatility series of Nokia on the Helsinki Stock Exchange around the burst of the IT-bubble. The essay is motivated by earlier findings which suggest that different scaling laws may apply to intraday time-scales and to larger time-scales, implying that the so-called annualized volatility depends on the data sampling frequency. The empirical results confirm the appearance of time varying long-memory and different scaling laws that, for a significant part, can be attributed to investor irrationality and to an intraday volatility periodicity called the New York effect. The findings have potentially important consequences for options pricing and risk management that commonly assume constant memory and scaling. The second essay investigates modelling the duration between trades in stock markets. Durations convoy information about investor intentions and provide an alternative view at volatility. Generalizations of standard autoregressive conditional duration (ACD) models are developed to meet needs observed in previous applications of the standard models. According to the empirical results based on data of actively traded stocks on the New York Stock Exchange and the Helsinki Stock Exchange the proposed generalization clearly outperforms the standard models and also performs well in comparison to another recently proposed alternative to the standard models. The distribution used to derive the generalization may also prove valuable in other areas of risk management. The third essay studies empirically the effect of decimalization on volatility and market microstructure noise. Decimalization refers to the change from fractional pricing to decimal pricing and it was carried out on the New York Stock Exchange in January, 2001. The methods used here are more accurate than in the earlier studies and put more weight on market microstructure. The main result is that decimalization decreased observed volatility by reducing noise variance especially for the highly active stocks. The results help risk management and market mechanism designing.