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Hopes for fractals

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Multifractal time as trading time
Benoît Mandelbrot Mathematician
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Now I would like to mention this matter of multifractal time as trading time because it is so fundamental. Again, the criticism that I heard of Bachelier's model of constant volatility in time was that it didn't take account of what is observed by just either watching or listening or reading newspapers - that some days the market is asleep, not much is happening, things drift up and down because somebody is getting married or somebody has died and they just have to buy and sell but nothing is happening really, a kind of background noise is ruling the roost. And on other days, an extraordinarily turbulent stock market, headlines on the front page of newspapers: the market went up or down by a huge amount, fortunes are made, fortunes are lost, etc., etc. The intuitive feeling that is in the minds of many people in the stock market is that, in a certain sense, the market moves fast or slowly. Now the implementation of it could have been to have a model of rapid and slow variation that would not be itself fractal, which would have derivatives, rates of variation. Well, that might have been tried. It was not tried, I think, very seriously. Whatever the case is, there is a very tempting way of introducing fractality, which means scaling variances throughout, which is by including in the definition of that time this property of scaling. And that is what I did. A first attempt actually came very early in the '60s, in a paper together with Taylor, in which I showed that if the trading time was moving in a certain fashion, by jumps and jumps and jumps, my early model of '63 would be resumed and reapplied. But immediately it was clear that this fractal time, as it was called, was very primitive and something richer was needed. Multifractal time fills this gap. Now in this model a very strong assumption is made, that the Brownian motion or fraction Brownian motion is independent of trading time. That is a very strong assumption. I'm very surprised that this strong assumption does not immediately hit some kind of difficulty because I don't think they could be independent. But, again, this is a matter of method. You make the simple assumption and you see how far it goes. You don't agonise by saying, "I'm over simplifying. I'm doing it wrong. I should be more careful. I should be more sophisticated. I should listen to more information." You just see how much a simple model contains without you knowing it. Therefore all this modelling is based very fundamentally on a distinction between, how should I say, traditional models and fractal modelling. By that I mean the following distinction that I could have made in other contexts, but which I might as well made here. When I was taught about modelling in science it was clear that if I have a simple model and simple construction; there is simple behaviour. If I want to get complicated behaviour I must have a complicated construction. This was sort of obvious. You look at prices; you hear it is complicated: I must have a complicated construction, a priori, a simple construction just could not do it. What is true of fractals throughout is the opposite. You start with a very simple construction and you get a behaviour that seems to be extraordinarily complicated, it is not complicated because you put in a great deal of rules, because what looks complicated, in fact, obeys a great deal of underlying deep order. And so, before resorting to further wrinkles, to further tuning of models, it is imperative to see how far a certain model would go, and then take this model as a basis for further building. If you build starting from Brownian motion, you don't restart until a pyramid of variable volatility, of discontinuities of everything, is built, and then you must still add to it. In my model you get variable volatility, discontinuities, everything, at will, from a simple formula. As long as it works it must not be criticised a priori; only criticised on its level of success.

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: 4 minutes, 54 seconds

Date story recorded: May 1998

Date story went live: 29 September 2010