Pointwise Regularity Exponents and Well-Behaved Residuals in Stock Markets

The article deals with a class of stochastic processes, the Multifractional Processes with Random Exponent (MPRE), recently introduced to gain flexibility in modeling many complex phenomena. We claim that MPRE can capture in a very parsimonious way most of the well known financial stylized facts. In particular, we prove that the process unconditional distributions are fat-tailed and high-peaked and show that, as it occurs for asset returns, the empirical autocorrelation functions of the process increments are close to zero whereas significant values are exhibited by squared (or absolute) increments. Furthermore, we provide evidence that the sole knowledge of functional parameter of the MPRE allows to calculate residuals that perform much better than those obtained by other discrete models such as the GARCH family