Lévy scaling in random walks with fluctuating variance

Abstract

Truncated Lévy flights with correlated fluctuations of the variance (heteroskedasticity) are considered. A stylized model is introduced, in which the variance fluctuates between two possible values following a Markov chain process. Analogously to conventional truncated Lévy flights with fixed variance, the central part of the probability distribution function of the increments at short time scales is found to be close to a Lévy distribution. What makes these processes interesting is the fact that the crossover to the Gaussian regime may occur for times considerably larger than for uncorrelated (or no) variance fluctuations. Processes of this type may find direct application in the modeling of some economic time series, in which Lévy scaling and heteroskedasticity are known to coexist.