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Multifractal time as trading time


The ability of the model to make predictions
Benoît Mandelbrot Mathematician
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[Q] Is your model able to make some predictions?

Well, predictions of various kinds. First of all, if the model is simplest, for example, Brownian motion of multifractal time, it will be a martingale, and therefore would not predict noticeable change because possibly there would be no change on the average. On the other hand, to say there is no change of the average is not enough. It is a very important prediction to say no change on the average, but the scatter will be small or big, and that the model can predict, not because I put it in, but because of the nature of the model. And I would like to emphasise the contrast between that and other models that were introduced, they are called GARCH, in which the variability of volatility is put in by hand, and therefore is found there at the exit, but only on a short run. In this model the variability is there very strongly, because of the assumptions made, and you can predict the variation, therefore, of the average. Secondly, when I say Brownian motion of multifractal time, I took the simplest case because the full model is fractional Brownian motion of multifractal time. After the fact, many of my friends said, 'Why did it take you so long to combine these two ideas? You've written a hundred papers on fractional Brownian motion, a hundred papers on multifractals: why didn't you immediately write a paper on fractional Brownian motion and multifractals?' My answer to them is very simple: 'Why didn't you do so?' It is a very strange thing how it is very difficult to think of which combination of existing tricks would do something, but in our testing of the Dollar/Deutschmark exchange we find that a certain key parameter is close to one half; one half is Brownian motion of multifractal time. But what you get is not quite one half. If it is not one half, then there is a prediction also on the direction of change. And very simply that can do it; it's a fundamental feature of it. How much more of prediction have we obtained? Well, I know a few more examples but I don't know everything. One must understand that my old model has been around for many years, has been knocked around, criticised, apparently demolished many times but always by totally non-lethal blows, but has been around and has been discussed. The new model is very, very recent. I feel it closes a loop in my thinking very strongly, and I'm very happy about it, but it has not been greatly discussed. So, discussion will elicit many more things about it one way or another.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Bernard Sapoval Daniel Zajdenweber

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Duration: 3 minutes

Date story recorded: May 1998

Date story went live: 29 September 2010