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Homogeneously Saturated Model for Development in Time of the Price of an Asset

The time development of the price of a financial asset is considered by constructing and solving Langevin equations for a homogeneously saturated model, and for comparison, for a standard model and for a logistic model. The homogeneously saturated model uses coupled rate equations for the money supply and for the price of the asset, similar to the coupled rate equations for population inversion and power density in a simple model of a homogeneously broadened laser. Predictions of the models are compared for random numbers drawn from a Student's t-distribution. It is known that daily returns of the DJIA and S&P 500 indices are fat tailed and are described well by Student's t-distributions over the range of observed values. The homogeneously saturated model shows returns that are consistent with daily returns for the indices (in the range of -30% to +30%) whereas the standard model and the logistic model show returns that are far from consistent with observed daily returns for the indices.