Introduction The concepts of scale invariance and scaling behavior are now increasingly applied outside their traditional domains of application, the physical sciences. Their application to financial markets, initiated by Mandelbrot [1; 2] in the 1960s, has experienced a regain of interest in the recent years, partly due to the abundance of high-frequency data sets and availability of computers for analyzing their statistical properties. This lecture is intended as an introduction and a brief review of current research in a field which is becoming increasingly popular in the theoretical physics community. We will try to show how the concepts of scale invariance and scaling behavior may be usefully applied in the framework of a statistical approach to the study of financial data, pointing out at the same time the limits of such an approach. 2 Rama CONT, Marc POTTERS & Jean-Philippe BOUCHAUD 2. Statistical description of market data

We define a large class of multifractal random measures and processes with arbitrary loginfinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk processes (MRW) [33, 3] and the log-Poisson “product of cynlindrical pulses” [7]. Their construction involves some “continuous stochastic multiplication” [36] from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non degeneracy, convergence of the moments and multifractal scaling.

Andrew Chou Jeremy Cooperstock y Ran El--Yaniv z Michael Klugerman x Tom Leighton -- November, 1993 Abstract The distributional approach and competitive analysis have traditionally been used for the design and analysis of on-line algorithms. The former assumes a specific distribution on inputs, while the latter assumes inputs are chosen by an unrestricted adversary. This paper employs the statistical adversary (recently proposed by Raghavan) to analyze and design on-line algorithms for two-way currency trading. The statistical adversary approach may be viewed as a hybrid of the distributional approach and competitive analysis. By statistical adversary, we mean an adversary that generates input sequences, where each sequence must satisfy certain general statistical properties. The on-line algorithms presented in this paper have some very attractive properties. For instance, the algorithms are money-making; they are guaranteed to be profitable when the optimal off-li...

This paper presents the multifractal model of asset returns (“MMAR”), based upon

Presented in an opening lecture of the XXXIXth

This paper considers regression models for cross-section data that exhibit crosssection dependence due to common shocks, such as macroeconomic shocks. The paper analyzes the properties of least squares (LS) estimators in this context. The results of the paper allow for any form of cross-section dependence and heterogeneity across population units. The probability limits of the LS estimators are determined, and necessary and sufficient conditions are given for consistency. The asymptotic distributions of the estimators are found to be mixed normal after recentering and scaling. The t� Wald, and F statistics are found to have asymptotic standard normal, χ2,andscaledχ2 distributions, respectively, under the null hypothesis when the conditions required for consistency of the parameter under test hold. However, the absolute values of t, Wald, and F statistics are found to diverge to infinity under the null hypothesis when these conditions fail. Confidence intervals exhibit similarly dichotomous behavior. Hence, common shocks are found to be innocuous in some circumstances, but quite problematic in others. Models with factor structures for errors and regressors are considered. Using the general results, conditions are determined under which consistency of the LS estimators holds and fails in models with factor structures. The results are extended to cover heterogeneous and functional factor structures in which common factors have different impacts on different population units.

c ○ 2006 by Elisa Luciano and Wim Schoutens. Any opinions expressed here are those of the authors and not those of the Collegio Carlo Alberto. We discuss a Lévy multivariate model for financial assets which incorporates jumps, skewness, kurtosis and stochastic volatility. We use it to describe the behavior of a series of stocks or indexes and to study a multi-firm, value-based default model. Starting from an independent Brownian world, we introduce jumps and other deviations from normality, including non-Gaussian dependence. We use a stochastic time-change technique and provide the details for a Gamma change. The main feature of the model is the fact that- opposite to other, non jointly Gaussian settings- its risk neutral dependence can be calibrated from univariate derivative prices, providing a surprisingly good fit.

We propose a method that detects the true direction of time series, by fitting an autoregressive moving average model to the data. Whenever the noise is independent of the previous samples for one ordering of the observations, but dependent for the opposite ordering, we infer the former direction to be the true one. We prove that our method works in the population case as long as the noise of the process is not normally distributed (for the latter case, the direction is not identifiable). A new and important implication of our result is that it confirms a fundamental conjecture in causal reasoning — if after regression the noise is independent of signal for one direction and dependent for the other, then the former represents the true causal direction — in the case of time series. We test our approach on two types of data: simulated data sets conforming to our modeling assumptions, and real world EEG time series. Our method makes a decision for a significant fraction of both data sets, and these decisions are mostly correct. For real world data, our approach outperforms alternative solutions to the problem of time direction recovery. 1.

This paper discusses two optimal allocation problems. We consider different hypotheses of portfolio selection with stable distributed returns for each of them. In particular, we study the optimal allocation between a riskless return and risky stable distributed returns. Furthermore, we examine and compare the optimal allocation obtained with the Gaussian and the stable non-Gaussian distributional assumption for the risky return. KEY WORDS: optimal allocation, stochastic dominance, risk aversion, measure of risk, a stable distribution, domain of attraction, sub-Gaussian stable distributed, fund separation, normal distribution, mean variance analysis, safety-first analysis. 2 1. INTRODUCTION This paper serves a twofold objective: to compare the normal with the stable non-Gaussian distributional assumption when the optimal portfolio is to be chosen and to propose stable models for the optimal portfolio selection according to the utility theory under uncertainty. It is well-known that asset returns are not normally distributed, but many of the concepts in theoretical and empirical finance developed over the past decades rest upon the assumption that asset returns follow a normal distribution. The fundamental work of Mandelbrot (1963a-b, 1967a-b) and Fama (1963,1965a-b) has sparked considerable interest in studying the empirical distribution of financial assets. The excess kurtosis found in Mandelbrot's and Fama's investigations led them to reject the normal assumption and to propose the stable Paretian distribution as a statistical model for asset returns. The Fama and Mandelbrot's conjecture was supported by numerous empirical investigations in the subsequent years, (see Mittnik, Rachev and Paolella (1997) and Rachev and Mittnik (2000)). The practical and theoretical app...

Summary. We propose two kernel based methods for detecting the time direction in empirical time series. First we apply a Support Vector Machine on the finitedimensional distributions of the time series (classification method) by embedding these distributions into a Reproducing Kernel Hilbert Space. For the ARMA method we fit the observed data with an autoregressive moving average process and test whether the regression residuals are statistically independent of the past values. Whenever the dependence in one direction is significantly weaker than in the other we infer the former to be the true one. Both approaches were able to detect the direction of the true generating model for simulated data sets. We also applied our tests to a large number of real world time series. The ARMA method made a decision for a significant fraction of them, in which it was mostly correct, while the classification method did not perform as well, but still exceeded chance level.