Centralizers of Subsystems of Fusion Systems

When $(S,\mathcal{F},\mathcal{L})$ is a $p$-local finite group and (T,\mathcal{E},\mathcal{\L}_0) is weakly normal in $(S,\mathcal{F},\mathcal{L})$ we show that a definition of $C_S(\mathcal{E})$ given by Aschbacher has a simple interpretation from which one can deduce existence and strong closure very easily. We also appeal to a result of Gross to give a new proof that there is a unique fusion system $C_{\mathcal{F}}(\mathcal{E})$ on $C_S(\mathcal{E})$.Comment: 9 page

A Multivariate Time-Changed Lévy Model for Financial Applications

The purpose of this paper is to define a bivariate L´evy process by subordination of a Brownian motion. In particular we investigate a generalization of the bivariate Variance Gamma process proposed in Luciano and Schoutens [8] as a price process. Our main contribution here is to introduce a bivariate subordinator with correlated Gamma margins. We characterize the process and study its dependence structure. At the end wealso propose an exponential Lévy price model based on our process.Levy processes, multivariate subordinators, dependence, multivariate asset modelling.

Tournaments, 4-uniform hypergraphs, and an exact extremal result

We consider $4$-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of $5$ vertices spans either $0$ or exactly $2$ hyperedges and give a construction, using quadratic residues, for an infinite family of such hypergraphs with the maximum number of hyperedges. Baber has previously given an asymptotically best-possible result using random tournaments. We give a connection between Baber's result and our construction via Paley tournaments and investigate a `switching' operation on tournaments that preserves hypergraphs arising from this construction.Comment: 23 pages, 6 figure

Centralizers of normal subgroups and the Z∗Z^*-Theorem

Glauberman's $Z^*$-theorem and analogous statements for odd primes show that, for any prime $p$ and any finite group $G$ with Sylow $p$-subgroup $S$, the centre of $G/O_{p^\prime}(G)$ is determined by the fusion system $\mathcal{F}_S(G)$. Building on these results we show a statement that seems a priori more general: For any normal subgroup $H$ of $G$ with $O_{p^\prime}(H)=1$, the centralizer $C_S(H)$ is expressed in terms of the fusion system $\mathcal{F}_S(H)$ and its normal subsystem induced by $H$.Comment: 3 pages; to appear in the Journal of Algebr

Trees of Fusion Systems

We define a `tree of fusion systems' and give a sufficient condition for its completion to be saturated. We apply this result to enlarge an arbitrary fusion system by extending the automorphism groups of certain of its subgroups

A Generalized Normal Mean Variance Mixture for Return Processes in Finance

Time-changed Brownian motions are extensively applied as mathematical models for asset returns in Finance. Time change is interpreted as a switch to trade-related business time, different from calendar time. Time-changed Brownian motions can be generated by infinite divisible normal mixtures. The standard multivariate normal mean variance mixtures assume a common mixing variable. This corresponds to a multidimensional return process with a unique change of time for all assets under exam. The economic counterpart is uniqueness of trade or business time, which is not in line with empirical evidence. In this paper we propose a new multivariate definition of normal mean-variance mixtures with a flexible dependence structure, based on the economic intuition of both a common and an idiosyncratic component of business time. We analyze both the distribution and the related process. We use the above construction to introduce a multivariate generalized hyperbolic process with generalized hyperbolic margins. We conclude with a stock market example to show the ease of calibration of the model.multivariate normal mean variance mixtures, multivariate generalized hyperbolic distributions, Levy processes, multivariate subordinators

Multivariate Variance Gamma and Gaussian dependence: a study with copulas

This paper explores the dynamic dependence properties of a Levy process, the Variance Gamma, which has non Gaussian marginal features and non Gaussian dependence. In a static context, such a non Gaussian dependence should be represented via copulas. Copulas, however, are not able to capture the dynamics of dependence. By computing the distance between the Gaussian copula and the actual one, we show that even a non Gaussian process, such as the Variance Gamma, can "converge" to linear dependence over time. Empirical versions of different dependence measures confirm the result.

Refinement Derivatives and Values of Games

A definition of set-wise differentiability for set functions is given through refining the partitions of sets. Such a construction is closely related to the one proposed by Rosenmuller (1977) as well as that studied by Epstein (1999) and Epstein and Marinacci (2001). We present several classes of TU games which are differentiable and study differentiation rules. The last part of the paper applies refinement derivatives to the calculation of value of games. Following Hart and Mas-Colell (1989), we define a value operator through the derivative of the potential of the game. We show that this operator is a truly value when restricted to some appropriate spaces of games. We present two alternative spaces where this occurs: the spaces pM( ) and POT2. The latter space is closely related to Myerson's balanced contribution axiom.TU games; large games; non-additive set functions; value; derivatives