Since the publications of Kolmogorov in the 1940s, the dominant approach has been to model turbulence using fractional or multi-fractional processes.
The experimental confirmation of the predictions made possible by the models has always, however, been a delicate matter.
Recently, the observation of time series taken from commodities, or currency, markets has made it possible to demonstrate in a direct way such multi-fractional behavior.
The Cournot Centre’s Probabilism research programme for 2018–19 is organized around two questions stemming from these results:
- On what kind of data is this type of behavior observable?
- Which micro-models are capable of taking into account these macroscopic processes?
In all of these fields of study, the nature of the data conditions the development of the models to be estimated. They have to be sampled correctly in order to allow for time-frequency analysis, and must not be filtered or pretreated. In any case, the fact that the processes are of a fractional or multi-fractional nature is fundamental. If modelling that uses Markov processes has been the standard stochastic approach, the multiplication of available data confirms that it is not only possible but necessary to go beyond this framework and develop new tools. The development of non-Markovian models would make possible the analysis of memory processes.
The concept of anomalous diffusion first refers to the case when the diffusion of a particle obeys a mean square displacement that is a power law t2H as a function of time t. H=1/2 corresponds to standard diffusion, or a random walk, as observed by Brown (context of Brownian motion) , while H<1/2 is a sub-diffusion, and H>1/2 a super-diffusion. Anomalous diffusion has been observed for inter-cell diffusion phenomena in biology, for instance. Markov processes, such as Levy processes, have been proposed for modelling such phenomena, but a more detailed analysis involving the correlations between increments has revealed memory effects.
Such problems with non-Markovian behavior are challenging to describe due to the non-local nature of the associated description of the probability density function. The process can then be modelled by a fractional or multifractional Brownian motion with the super-diffusive case (H>1/2) corresponding to a persistent process and sub-diffusion (H<1/2) with anti-persistence.
There are, moreover, important challenges associated with modelling, simulation and analysis of such phenomena when trying to understand them through comparison with experimental data. Non-Markovian models may, in particular, be important for modelling phenomena that come about as a result of human intervention; indeed, this may mean that there is “memory” in the process leading to a non-Markovian behaviour as, for instance, recently seen in price processes.