A reflexive toy-model for financial market

We propose a reflexive toy model for market dynamics, based on the idea that existing reflexive loops are generated by the conviction, shared by many market operators, that a certain price follows a certain model. Their trading behaviour will therefore increase the probability that the model predictions are in fact fulfilled. We analytically write the equations generating a reflexive loop stemming from a simple linear regression model, and we show that the resulting toy model yields a peculiar intermittent behavior. The presence of two unstable fixed points is apparent from our numerical calculation and the residence-time distribution density in these points asymptotically follows an inverse-power-law tail. The exponent of this tail, as well as the scaling properties of the model output, are close to those stemming from real-price time series.