Distributions of historic market data: relaxation and correlations

We investigate relaxation and correlations in a class of mean-reverting models for stochastic variances. derive closed-form expressions the correlation functions leverage general form term. also discuss three specific models— multiplicative, Heston (Cox-Ingersoll-Ross) combined multiplicative-Heston—whose steady-state probability density are Gamma, Inverse Gamma Beta Prime respectively, latter two exhibiting “fat” tails. For model, we apply eigenvalue analysis Fokker-Planck equation to function—in agreement with analysis— identify series time scales, which observable cumulants on approach steady state. test our findings very large set historic financial markets data.