  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. We predict that the shape
of the falldown of the price during a crash should be logarithmic. 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.
