Chaotic time series applied to finance statistical and algorithmic problems.

Summary Our contribution to the study of chaotic time series focuses on the following points. We propose a study framework allowing to take into account the contributions of the various scientific fields dealing with this type of data: signal processing, dynamical systems, ergodic theory, finance and statistics. We clarify the notion of global Lyapunov exponent with the help of several definitions, from the most formal to the most commonly used. We show why global Lyapunov exponents are useful for characterizing chaos but practically useless for forecasting. We then focus on local Lyapunov exponents. We show how each definition is related to the global Lyapunov exponents and we specify in which application framework each definition is relevant. We give a new result concerning the distribution of local Lyapunov exponents. We consider the non-parametric prediction methods that can be used in this context, detailing two that seem particularly adapted to chaos: nearest neighbors and radial functions. The latter estimator is studied in more detail. We specify its properties and give an algorithm to implement it. We study the predictability of chaotic time series. We show how the prediction horizon is related to the local Lyapunov exponents of the system. We propose a new theoretical approach to deal with the case of noisy chaos. We address the problem of choosing a sampling step for chaotic series from a continuous time system. We give a new result allowing to choose an optimal sampling step in the sense of the prediction horizon. We support these presentations with a set of simulations from known chaotics by specifying their algorithmic costs. We discuss the problems posed by the simulation of chaotic time series. Finally, we give two applications of the tools developed in the framework of intraday financial series. The first application is a direct illustration of these tools in the case of exchange rates. The second application makes prior use of time warping methods which are presented here in a new unified framework.