Fluctuating confidence and stock-market returns

The drift of two different diffusion processes (asset returns) is determined by a state variable which can take on two values. It jumps between the two according to Poisson increments (this is called a 'regime-switch'). For any given position of the state variable the drift of one process is high and the other is low. I find that the posterior probability that the 1st asset has higher average returns, conditional on observing the path (returns) of each process, follows a diffusion process and calculate its infinitesimal parameters. I also derive analytical expressions for its stationary density and for some of its path properties. I compare the filtering problem to the Kalman Filtering problem and find that even though the dynamics of the mean of the distribution are very similar, the dynamics of the variance are subject to stochastic fluctuations. The model is parsimonious in that the conditional mean and variance are functions of a single variable. ; I characterize the interest-rate and total-returns processes in a Cox-Ingersoll-Ross[1985] style model where the productivities of assets are unobserved, but inferred as above. I find that this model is capable of reproducing three stylized facts of stock-market returns and interest-rates. These are the skewness and kurtosis of returns and the 'Predictive-Asymmetry' of returns: excess-returns and future changes in volatility are negatively correlated. Further negative returns cause reactions of larger magnitude. The success of the model in generating these features depends on the speed of learning about the regime switches. Parameter values which lead to faster learning, are consistent with large negative skewness of returns and the Predictive Asymmetry property. The slower learning version leads to greater kurtosis of returns. I show that a model based on the same fundamentals but with observed 'regime-shifts' is not reconcilable with these features. My analysis suggests that learning about the productivities of assets of the kind introduced here may be an important determinant of portfolio choices and observed asset returns.