Arthur Matsuo Yamashita Rios de Sousa, Hideki Takayasu, Misako Takayasu

We study autoregressive process (AR) based models for financial time series, focusing on their diffusion behavior. First, we solve the basic AR process, deriving the analytical diffusion expression for arbitrary order and obtaining the conditions on the AR coefficients for the process to exhibit ordinary diffusion (mean square displacement linear in time) for large time scale but abnormal diffusion for short time scale. Then, we analyze the Potential of Unbalanced Complex Kinetics (PUCK) model, which describes the position of a random walker (or prices) subjected to a potential centered in its own past moving average. For a quadratic potential, this model reduces to a particular case of the AR process and the previously obtained results apply. Next, we develop the AR process with time-dependent coefficients and identify a special case of random correlated coefficients with a PUCK model in which the quadratic potential fluctuates; by adding random fluctuations to the original fixed potential of the PUCK model, we have a power-law distribution for price differences but keeping the same diffusion behavior. At last, we characterize real data from the foreign exchange market using the fluctuating potential PUCK model.