Abstract: Reams and reams have been written in quantitative finance about the unsolved problem of the stock markets efficiency. Starting from the seminal work of Fama (1970), who defined the Efficient Market Hypothesis in terms of expected values conditional to the informative set ("financial assets fully reflect all available information that is relevant to their values"), a huge number of works have tried to address the question whether real financial markets behave efficiently. Roughly speaking, the EMH argues that market does price assets broadly correctly, excluding that deviations from equilibrium values could last for long. So great was the consensus met by the EMH to set one of the most impressive bodies of knowledge of the 20th century: the mathematical finance.
Nonetheless, the number of equilibrium theorems proved under the assumptions of the EMH grew at the same rate of the empirical evidence that made questionable the validity of the EMH itself.
Many approaches have been followed in literature to test the EMH. In spite of all the efforts, to date the problem remains open and current more than ever, basically because of two main reasons:

- the analyses, particularly those aiming at testing the random walk model in its different specifications, often provide non conclusive results;
- the real world dynamics, with their repeated financial crises made of bubbles and crashes, seriously do challenge the credibility of the EMH, to the extent that a strand of skeptical thought, the behavioural finance, has been booming. One of the most quoted works in this context is due to DeBondt and Thaler (1985); they show that using historical returns abnormal profits are achievable in the long-run, simply going short a portfolio of 'winner stocks' (i.e., stocks with good performances in the past) and going long a portfolio of 'loser stocks' (i.e., stocks that performed badly in the past). What causes these opposite profits, according to the authors, is investors' excess of optimism and pessimism, the so called overreaction to information. Following this analysis, a plethora of contributions provided evidence of reverse abnormal profits and documented them in international markets and for short time horizons. Other empirical results suggest that prices underreact to information in a way to generate the so called the "momentum" profit; the trading strategy in this case consists in going long a portfolio of extremely winner stocks in the past and going short a portfolio of extremely loser stocks.

Results like those just recalled raise the question whether a model exists able to make consistent all these opposite findings. The paper concludes in favour of an affirmative answer: a model is discussed that, recently defined by Ayache and Taqqu (2005) in a general setting, succeeds in reconciling efficiency and behavioural finance. The model, named Multifractional Processes with Random Exponent (MPRE), emanates from the well-known fractional Brownian motion (fBm), the unique Gaussian process, self-similar of order H, vanishing at the origin with stationary increments, introduced to model long-range dependence. The parameter H of the fBm is informative of the intensity of dependence and of the regularity of the process' paths. Yet, the constancy of H is undesirable for many phenomena, whose complexity requires the pointwise regularity to change over time, even abruptly. When H is replaced by a random function one gets the MPRE. The construction of this class of processes considers (a) a Gaussian field depending on H and the time domain, and (b) a random variable or a stochastic process with values in an arbitrary fixed compact interval. The resulting process is versatile enough to describe very complex phenomena such as stock price dynamics.

Brief Biography of the Speaker:
Sergio Bianchi, born on 25 September 1967, graduated in 1991 from University of Cassino (Italy) in Economics. He earned his Ph.D. in Actuarial Science from University of Rome "La Sapienza" (Italy). After teaching experiences as invited professor at the Pontifical Gregorian University (Vatican State) and at University of Sassari (Italy), in 1998 was appointed assistant professor at University of Cassino, where became associate professor in 2001 and full professor of Mathematics and Financial Mathematics in 2006. Since 2003 to 2005 he was scientific responsible for the Computer Laboratory of the Department "Economia e Territorio" at University of Cassino. He also held the office of Head of the Department "Istituzioni, Metodi Quantitativi e Territorio" from 2005 to 2009, when he resigned after the appointment as Rector's Delegate for Research and Benchmarking. Member of the board of professors of the Ph.D. in Economics at the same University, he is referee for a number of international journals and the author of more than fifty research papers concerning the modeling of financial time series by (multi)fractional processes and self-similar stochastic processes.