Texto completo, versión de autor.- PACS Nos.: 05.45.Tp, 02.50.Sk, 05.40.-a, 05.45.Ra, 89.75.FbThis article and the companion paper aim at reviewing recent empirical and theoretical developments usually grouped under the term Econophysics. Since the name was coined in 1995 by merging the words ‘Economics’ and ‘Physics’, this new interdisciplinary field has grown in various directions: theoretical macroeconomics (wealth distribution), microstructure of financial markets (order book modeling), econometrics of financial bubbles and crashes, etc. We discuss the interactions between Physics, Mathematics, Economics and Finance that led to the emergence of Econophysics. We then present empirical studies revealing the statistical properties of financial time series. We begin the presentation with the widely acknowledged ‘stylized facts’, which describe the returns of financial assets—fat tails, volatility clustering, autocorrelation, etc.—and recall that some of these properties are directly linked to the way ‘time’ is taken into account. We continue with the statistical properties observed on order books in financial markets. For the sake of illustrating this review, (nearly) all the stated facts are reproduced using our own high-frequency financial database. Finally, contributions to the study of correlations of assets such as random matrix theory and graph theory are presented. The companion paper will review models in Econophysics from the point of view of agent-based modeling.Peer reviewe

This article is the second part of a review of recent empirical and theoretical developments usually grouped under the heading Econophysics. In the first part, we reviewed the statistical properties of financial time series, the statistics exhibited in order books and discussed some studies of correlations of asset prices and returns. This second part deals with models in Econophysics from the point of view of agent-based modeling. Of the large number of multi-agent-based models, we have identified three representative areas. First, using previous work originally presented in the fields of behavioral finance and market microstructure theory, econophysicists have developed agent-based models of order-driven markets that we discuss extensively here. Second, kinetic theory models designed to explain certain empirical facts concerning wealth distribution are reviewed. Third, we briefly summarize game theory models by reviewing the now classic minority game and related problems.

We study a statistical model consisting of $N$ basic units which interact with each other by exchanging a physical entity, according to a given microscopic random law, depending on a parameter $\lambda$. We focus on the equilibrium or stationary distribution of the entity exchanged and verify through numerical fitting of the simulation data that the final form of the equilibrium distribution is that of a standard Gamma distribution. The model can be interpreted as a simple closed economy in which economic agents trade money and a saving criterion is fixed by the saving propensity $\lambda$. Alternatively, from the nature of the equilibrium distribution, we show that the model can also be interpreted as a perfect gas at an effective temperature $T\left(\lambda \right)$, where particles exchange energy in a space with an effective dimension $D\left(\lambda \right)$.

Anomalous transport in a tilted periodic potential is investigated numerically within the framework of the fractional Fokker-Planck dynamics via the underlying continuous-time random walk. An efficient numerical algorithm is developed which is applicable for an arbitrary potential. This algorithm is then applied to investigate the fractional current and the corresponding nonlinear mobility in different washboard potentials. Normal and fractional diffusion are compared through their time evolution of the probability density in state space. Moreover, we discuss the stationary probability density of the fractional current values.

We investigate a subdiffusive, fractional Fokker-Planck dynamics occurring in time-varying potential landscapes and thereby disclose the failure of the fractional Fokker-Planck equation (FFPE) in its commonly used form when generalized in an ad hoc manner to time-dependent forces. A modified FFPE (MFFPE) is rigorously derived, being valid for a family of dichotomously alternating force fields. This MFFPE is numerically validated for a rectangular time-dependent force with zero average bias. For this case, subdiffusion is shown to become enhanced as compared to the force free case. We question, however, the existence of any physically valid FFPE for arbitrary varying time-dependent fields that differ from this dichotomous varying family.