THE availability of high-frequency data for financial markets has made it possible to study market dynamics on timescales of less than a day 1 • For foreign exchange (FX) rates Miiller et al. 2 have shown that there is a net flow of information from long to short timescales: the behaviour of long-term traders (who watch the markets only from time to time) influences the behaviour of short-term traders (who watch the markets continuously). Motivated by this hierarchical feature, we have studied FX market

Stylized facts for univariate high-frequency data in finance are well known. They include scaling behaviour, volatility clustering, heavy tails and seasonalities. The multivariate problem, however, has scarcely been addressed up to now. In this paper, bivariate series of high-frequency FX spot data for major FX markets are investigated. First, as an indispensable prerequisite for further analysis, the problem of simultaneous deseasonalization of high-frequency data is addressed. In the following sections we analyse in detail the dependence structure as a function of the timescale. Particular emphasis is put on the tail behaviour, which is investigated by means of copulas.

We study the rate of irreversible entropy production and the entropy flux generated by low-dimensional dynamical systems modeling transport processes induced by the simultaneous presence of an external field and a density gradient. The key ingredient for understanding entropy balance is the coarse graining of the phasespace density. This mimics the fact that ever refining phase-space structures caused by chaotic dynamics can only be detected by finite resolution. Calculations are carried out for a generalized multibaker map. For the time-reversible dissipative ͑thermostated͒ version of the model, results of nonequilibrium thermodynamics are recovered in the large system limit. Independent of the choice of boundary conditions, we obtain the rate of irreversible entropy production per particle as u 2 /D, where u is the streaming velocity ͑current per density͒ and D is the diffusion coefficient. ͓S1063-651X͑98͒09407-0͔

A multibaker map is generalized in order to mimic the thermostating algorithm of transport models. Elementary calculations yield the irreversible entropy production caused by coarse graining of the phase-space density. For different systems, either in steady states (periodic or flux boundaries) or subjected to absorbing boundaries, the specific irreversible entropy production is shown to be u 2 ͞D, where u denotes the local streaming velocity (current per density) and D is the diffusion coefficient. [S0031-9007(97)04219-1] PACS numbers: 05.70.Ln, 05.45. + b, 51.10. + y The connection between nonequilibrium statistical physics and the underlying chaotic dynamics has become a subject of vivid interest [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The central questions are how the microscopic reversible dynamics can appear as an irreversible process on the macroscopic level, and how the macroscopic transport coefficients are related to microscopic characteristics of the chaotic dynamics. A careful analysis of the rate of irreversible entropy production is at the heart of this problem [16 -20], but the relation between complementary approaches has been poorly understood. Here, we present a consistent derivation of the irreversible entropy production for three main approaches to describe transport in driven systems producing currents. They model either nonequilibrium steady states (A) or the relaxation towards steady states (B):(A1) In the thermostating algorithm a special force is introduced to avoid an uncontrolled growth of the kinetic energy of particles moving in external fields [2][3][4][5][6][7][8][9]. The force mimics the presence of a thermostat and makes the particle dynamics dissipative on average, although it preserves time reversibility. The systems investigated up to now were assumed to be periodic of large spatial extension, and hence to be closed. The long time dynamics exhibits permanent chaos on an underlying chaotic attractor. Transport coefficients and the irreversible entropy production are connected with the average phase-space contraction rates ͑ p͒ on the attractor.(A2) By applying flux boundary conditions to an open Hamiltonian system it was shown that the steady-state density follows Fick's law [10], and the irreversible entropy production has been calculated [19]. In this case the current running through the system is due to the boundary condition only, and the phase-space contraction rate is zero,s ͑ f͒ 0.(B) The escape-rate formalism of transport processes is based on the investigation of open systems of large spatial extensions subjected to absorbing boundary conditions [11 -13]. In such cases the particle dynamics is chaotic in the sense of transient chaos, and there exists an underlying nonattracting chaotic set (a chaotic saddle) in the phase space. In the regime of linear response, at least, relaxation is closely related to steady-state transport. The transport coefficients of Hamiltonian systems are related [1,[11][12][13] to the chaotic saddle's escape rate k. In a recent ...

We review recent results concerning entropy balance in low-dimensional dynamical systems modeling mass (or charge) transport. The key ingredient for understanding entropy balance is the coarse graining of the local phase-space density. It mimics the fact that ever refining phase-space structures caused by chaotic dynamics can only be detected up to a finite resolution. In addition, we derive a new relation for the rate of irreversible entropy production in steady states of dynamical systems: It is proportional to the average growth rate of the local phase-space density. Previous results for the entropy production in steady states of thermostated systems without density gradients and of Hamiltonian systems with density gradients are recovered. As an extension we derive the entropy balance of dissipative systems with density gradients valid at any instant of time, not only in stationary states. We also find a condition for consistency with thermodynamics. A generalized multi-Baker map is used as an illustrative example. (c) 1998 American Institute of Physics.