This paper introduces an ARCH model (exponential ARCH) that (1) allows correlation between returns and volatility innovations (an important feature of stock market volatility changes), (2) eliminates the need for inequality constraints on parameters, and (3) allows for a straightforward interpretation of the "persistence" of shocks to volatility. In the above respects, it is an improvement over the widely-used GARCH model. The model is applied to study volatility changes and the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987. Copyright 1991 by The Econometric Society.

Conditional heteroskedasticity in asset returns: a new approach

In Weiterentwicklung fruherer Theorien von J. und F. Perrin und klassischphysikalischer Uberlegungen des Verfassers wird eine quantenmechanische Behandlung des Ubergangs von Elektronenanregungsenergie zwischen gleichartigen Molekulen in Losung gegeben. Der kritische Molekulabstand, unterhalb dessen der ubergang wahrend der Anregungsdauer stattfindet, last sich aus den Absorptions- und Fluoreszenzspektren und der Anregungsdauer der Molekule berechnen. Fur Fluorescein und Chlorophyll a ergeben sich Werte von 50 bzw. 80 AE, entsprechend den mittleren Molekulabstanden in Losungen von 3,2 · 10−3 bzw. 7,7 · 10−4 Molen/Liter. Fur die Bereiche oberhalb und unterhalb der kritischen Konzentration werden Formeln zur Berechnung der Energieabwanderung vom Primarmolekul angegeben, die mit den vorliegenden Messungen der Konzentrationsdepolarisation der Fluoreszenz gut ubereinstimmen. Die Anwendung auf analoge Energiewanderungsprobleme in Molekulkristallen und im Assimilationsapparat der Pflanze wird diskutiert.

Zwischenmolekulare Energiewanderung und Fluoreszenz

These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity.

http://cds.cern.ch/record/265648/files/9407018.pdf

Geometry of 2D topological field theories

We describe a method for determining chemical kinetic constants and diffusion coefficients by measuring the rates of decay of spontaneous concentration fluctuations. The equilibrium of the system is not disturbed during the measurement. We measure the number of molecules of a specified type in a defined open volume as a function of time and compute the time course of the deviations from the thermodynamic mean concentration. The method is based on the principle that the rates of decay of spontaneous microscopic fluctuations are determined by the same phenomenological rate coefficients as those of macroscopic departures from equilibrium which result from external perturbations. Hence, an analysis of fluctuations yields the same chemical rate constants and diffusion coefficients as are measured by conventional procedures. In practice the number of the specified molecules is measured by a property such as absorbance or fluorescence which is specific and sensitive to chemical change. The sample volume is defined by a light beam which traverses the cell. As the molecules appear in or disappear from the light beam, either due to diffusion or chemical reaction, their concentration fluctuations give rise to corresponding fluctuations of the intensity of absorbed or emitted light. This paper presents the theory needed to derive chemical rate constants and diffusion coefficients from these fluctuations in light intensity. The theory is applied to three examples of general interest: pure diffusion in the absence of chemical reaction; the binding of a small rapidly diffusing ligand to a larger slowly diffusing macromolecule; and a unimolecular isomerization. The method should be especially useful in studying highly cooperative systems, relatively noncooperative systems with intermediate states closely spaced in free energy, small systems, and systems not readily subject to perturbations of state.

Fluorescence correlation spectroscopy. I. Conceptual basis and theory

This paper establishes necessary and sufficient conditions for the stationarity and ergodicity of the GARCH(l.l) process. As a special case, it is shown that the IGARCH(1,1) process with no drift converges almost surely to zero, while IGARCH(1,1) with a positive drift is strictly stationary and ergodic. We examine the persistence of shocks to conditional variance in the GARCH(l.l) model, and show that whether these shocks "persist" or not depends crucially on the definition of persistence. We also develop necessary and sufficient conditions for the finiteness of absolute moments of any (including fractional) order.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0266466600005296

Stationarity and Persistence in the GARCH(1,1) Model

http://www.gbv.de/dms/ilmenau/toc/039945227.PDF

Formulas and Theorems for the Special Functions of Mathematical Physics

The present investigation was stimulated by a recent paper of K. 0. Friedrichs 113, who arrived at some purely algebraic problems in connection with the theory of linear operators in quantum mechanics. In particuIar, Friedrichs used a theorem by which the Lie elements in a free associative ring can be characterized. This theorem is proved in Section I1 of the present paper together with some applications which concern the addition theorem of the exponential function for non-commuting variables, the so-called BakerHausdorff formula. Section I contains some algebraic preliminaries. It is of a purely expository character and so is part of Section 111. Otherwise, Section 111 deals with the following problem, also considered by Friedrichs: Let A( t ) be a linear operator depending on a real variable 1. Let Y(t) be a second operator satisfying the differential equation

Wilhelm Magnus

Papers

Formulas and Theorems for the Special Functions of Mathematical Physics

On the exponential solution of differential equations for a linear operator

Combinatorial Group Theory.

Hill's equation

Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations