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## LIKELIHOOD RATIO TESTS FOR COMPETING FOKKER-PLANCK EQUATIONS FOR FINANCIAL RETURNS

Suppose you have $N$ daily returns $R = \{r_1, \dots, r_N\}$ and you have competing Fokker-Planck equations $(\partial_t + A_1) p(t,x) = 0$ and $(\partial_t + A_2) p(t,x)$, and wish to know which of the two Fokker-Planck equatons fit the data $R$ 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 $T=1,\dots,N$ 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 $A_1$ the Levy stable density with symbol (Fourier tranform multiplier) $\exp( -a |\xi|^b)$ and for $A_2$ 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 $0<\gamma<1$ by

$D^\gamma_t f(t) = \frac{1}{\Gamma(-\gamma)} \int_0^t (t-s)^{-\gamma-1} f(s) ds$

and these have the semigroup property under composition, i.e., $D^a_t D^b_t=D^{a+b}_t$ and the basic solution of $D^\gamma_t f(t) = \lambda f(t)$ is solved by $f(t) = E_{\gamma}(\lambda t^\gamma)$ (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) $d/dt P(t,\xi) + c |\xi|^{\alpha} P(t,\xi) = 0$ and (b) $D^\gamma_t Q(t,\xi) + c |\xi|^{\alpha} Q(t,\xi)=0$.  Note that the second is obtained from the non-Markovian Fokker-Planck equation by applying a fractional time derivative to it to eliminate that $A_2$.  Now both (a) and (b) can be solved analytically, the second by a Mittag-Leffler function.  So (a) is trivially solved by $P(t,\xi) = \exp( - c|\xi|^\alpha t)$ while (b) is solved by $Q(t,\xi) = E_{\gamma}( - c|\xi|^\alpha t^\gamma)$.