In this paper we present a rather general phenomenological theory of tick-by-tick dynamics in financial markets. Many well-known aspects, such as the Lévy scaling form, follow as particular cases of the theory. The theory fully takes into account the non-Markovian and non-local character of...

We introduce Bayesian estimation into the Tikhonov regularisation process for recovering functions of high bandwidth from noisy band-limited observations. The method allows for the incorporation of featural constraints and requires the global minimisation of a functional derived from damped...

We apply an asymmetric version of Kirman's herding model to volatile financial markets. In the relation between returns and agent concentration we use the square root law proposed by Zhang. This can be derived by extending the idea of a critical mean field theory suggested by Plerou et al. We...

We consider different levels of complexity which are observed in the empirical investigation of financial time series. We discuss recent empirical and theoretical work showing that statistical properties of financial time series are rather complex under several ways. Specifically, they are...

A symmetric phase field model is used to study wavelength selection in two dimensions. We study the problem in a finite system using a two-pronged approach. First we construct an action and, minimizing this, we obtain the most probable configuration of the system, which we identify with the...

We consider different models of stochastic dissipative equations and theoretically compute the probability distribution functions (actually the associated large deviation functions) of the time averaged injected power required to sustain a nontrivial stationary state. We discuss the results and...

We consider the processes defined by a Langevin equation and the associated continuity equation. The average of the density function, solution of the continuity equation, satisfies the Fokker–Planck equation. For a volume preserving vector field the same equation is satisfied by the average of...

We perform a parallel analysis of the spectral density of (i) the logarithm of price and (ii) the daily number of trades of a set of stocks traded in the New York Stock Exchange. The stocks are selected to be representative of a wide range of stock capitalization. The observed spectral densities...

In this paper we derive analytic formulae for statistical arbitrage trading where the security price follows an Ornstein–Uhlenbeck process. By framing the problem in terms of the first-passage time of the process, we derive expressions for the mean and variance of the trade length and the...

In these lectures we give an overview of nonequilibrium stochastic systems. In particular we discuss in detail two models, the asymmetric exclusion process and a ballistic reaction model, that illustrate many general features of nonequilibrium dynamics: for example coarsening dynamics and...