Random coefficient autoregressive processes and the PUCK model with fluctuating potential

Abstract

The potentials of unbalanced complex kinetics (PUCK) model consists of a random walker subjected to a potential centered at its moving average position. We study the PUCK model with fluctuating quadratic potential, showing that it is a special case of the random coefficient autoregressive (RCAR) process and thus a member of the same class of processes as the autoregressive conditional heteroskedasticity (ARCH) process; both RCAR processes but with different coefficient dependence. For the general Gaussian RCAR process, we use generalized Fibonacci numbers to derive explicit expressions for the mean squared displacement and the autocovariance function. We also obtain the conditions for divergence of variance, which imply tail index of the power-law-tailed cumulative probability distribution. Translating the results to the PUCK model, we analyze US Dollar/Japanese Yen market data and demonstrate that the model simultaneously reproduces empirical facts of price time series: diffusion and autocorrelation of prices and heavy-tailed distribution of price changes.