A Langevin approach to stock market fluctuations and crashes

We propose a non linear Langevin equation as a model for stock market fluctuations and crashes. This equation is based on an identification of the different processes influencing the demand and supply, and their mathematical transcription. We emphasize the importance of feedback effects of price variations onto themselves. Risk aversion, in particular, leads to an "up-down" symmetry breaking term which is responsible for crashes, where "panic" is self reinforcing. It is also responsible for the sudden collapse of speculative bubbles. Interestingly, these crashes appear as rare, "activated" events, and have an exponentially small probability of occurence. The model leads to a specific "shape" of the falldown of the price during a crash, which we compare with the October 1987 data. The normal regime, where the stock price exhibits behavior similar to that of a random walk, however reveals non trivial correlations on different time scales, in particular on the time scale over which operators perceive a change of trend.