Marie-Eliette Dury

Method for Model selection for Multifractal processes and some applications

Résumé

The multifractional Brownian motion (mBm) can be viewed as a generalization of the fractional Brownian motion (fBm) where the Hurst index H is replaced by a time-varying function H(t).

Fractional processes are encountered in different kind of applications.

For instance, arbitrage opportunity is possible when the Hurst index is constant and known in advance, but no more when the Hurst index is time-varying and even random. Moreover, a period with Hurst index significantly different from H=1/2 , specific case with independence of the increments, can be understood as follows: When H(t)<1/2, the market overreacts, namely in probabilistic term the increments of the (log)price process are negatively correlated (anti-persistence), whereas when H(t)>1/2, the market underreacts, namely the increments of the (log)price process are positively correlated (persistence). In behavioral economics, under-reaction can be viewed as overconfidence of investors.
For such a time-varying Hurst index, the methods of estimation developed up to now localize the estimation of Hurst index on small vicinity, for models that become more and more sophisticated, e.g. Hurst index being itself a stochastic process. The problem, with those new generalizations of mBm where a Hurst index which can be very irregular and even stochastic, is that we cannot know whether fluctuations reflect the reality or are just an artifact of the statistics.

The aim is here to provide the simplest possible model with a time-varying Hurst index. Such models should fit well the empirical Hurst index. This presentation describes an approach giving a method for the selection of a good probabilistic model with a time-varying Hurst index. The guiding idea, common thread of the test, is to choose a quite simple function H(t) which describes the real dataset as well as a more complicated one.

To sum up, the naive multifractional estimator of H(t) has too many fluctuations that appear as a statistical artefact. Then it should be asymptotically rejected by the test developed with this method. Moreover this propose a way to choose the simplest possible estimator H_n(t) to define an appropriated function H(t)tilde.

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