Suppose you have daily returns and you have competing Fokker-Planck equations and , and wish to know which of the two Fokker-Planck equatons fit the data better. For this problem, a direct likelihood computation is possible if both the Fokker-Planck equations can be simulated with the same initial conditions since at each time point the density can be evaluated at the data point, and the log-likelihood is thus simply the evaluation of the probability density function. For serious examples, we might want to take for the Levy stable density with symbol (Fourier tranform multiplier) and for a Levy stable distribution with a fractional time derivative composed with it, which leads to the Fokker-Planck equation with a memory kernel (since Riemann-Liouville time-fractional integral is simply comvolution with fractional time power).

Recall that the Caputo fractional time derivative is defined for by

and these have the semigroup property under composition, i.e., and the basic solution of is solved by (see e.g. (10.6) of mittag-leffler-apps-2009).

MARKOVIAN AND NON-MARKOVIAN FOKKER-PLANCK EQUATIONS FOR FINANCIAL RETURNS

We consider the Fourier-transformed forms. (a) and (b) . Note that the second is obtained from the non-Markovian Fokker-Planck equation by applying a fractional time derivative to it to eliminate that . Now both (a) and (b) can be solved analytically, the second by a Mittag-Leffler function. So (a) is trivially solved by while (b) is solved by .

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