A relaxation scheme for computation of the joint spectral radius of matrix sets

The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.Comment: 16 pages, 2 figures, corrected typos, accepted for publication in JDE

Periodic sequences of arbitrage: a tale of four currencies

This paper investigates arbitrage chains involving four currencies and four foreign exchange trader‐arbitrageurs. In contrast with the three‐currency case, we find that arbitrage operations when four currencies are present may appear periodic in nature, and not involve smooth convergence to a ‘balanced’ ensemble of exchange rates in which the law of one price holds. The goal of this article is to understand some interesting features of sequences of arbitrage operations, features which might well be relevant in other contexts in finance and economics

Double Exponential Instability of Triangular Arbitrage Systems

If financial markets displayed the informational efficiency postulated in the efficient markets hypothesis (EMH), arbitrage operations would be self-extinguishing. The present paper considers arbitrage sequences in foreign exchange (FX) markets, in which trading platforms and information are fragmented. In Kozyakin et al. (2010) and Cross et al. (2012) it was shown that sequences of triangular arbitrage operations in FX markets containing 4 currencies and trader-arbitrageurs tend to display periodicity or grow exponentially rather than being self-extinguishing. This paper extends the analysis to 5 or higher-order currency worlds. The key findings are that in a 5-currency world arbitrage sequences may also follow an exponential law as well as display periodicity, but that in higher-order currency worlds a double exponential law may additionally apply. There is an "inheritance of instability" in the higher-order currency worlds. Profitable arbitrage operations are thus endemic rather that displaying the self-extinguishing properties implied by the EMH.Comment: 22 pages, 22 bibliography references, expanded Introduction and Conclusion, added bibliohraphy reference

Discretized rotation has infinitely many periodic orbits

For a fixed k in (-2,2), the discretized rotation on Z^2 is defined by (x,y)->(y,-[x+ky]). We prove that this dynamics has infinitely many periodic orbits.Comment: Revised after referee reports, and added a quantitative statemen

The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields

We consider an "elastic" version of the statistical mechanical monomer-dimer problem on the n-dimensional integer lattice. Our setting includes the classical "rigid" formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan" EDM systems where the dimer potential is a weighted l1-distance and the auxiliary GRF is a Markov random field of Pickard type which behaves in space like autoregressive processes do in time. For one-dimensional Manhattan EDM systems, we compute the MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a compact transfer operator on a Hilbert space which is related to the annihilation and creation operators of the quantum harmonic oscillator and also recast it as the eigenvalue problem for a pantograph functional-differential equation.Comment: 24 pages, 4 figures, submitted on 14 October 2011 to a special issue of DCDS-