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## GREY BROWNIAN MOTIONS AS MODELS AS LINEAR PART OF FINANCIAL RETURNS

Schneider’s grey Brownian motions are two parameter stochastic processes $B_{\alpha,\beta}(t)$ starting at zero with variance $E(B_{\alpha,\beta}^2(t))=\frac{2}{\Gamma(\beta+1)} t^\alpha$ and characteristic function $E_{\beta}(-y^2 |t-s|^\alpha)$.  The marginal density of the gBM is given by

$f_{\alpha,\beta}(t,x) =\frac{t^{-\alpha/2}}{\sqrt{2}} M_{\beta/2}(\sqrt{2}|x|t^{-\alpha/2})$ (*)

where $M_{\beta}$ is the M-Wright function with Laplace transform $E_\beta(-s)$.  Now the marginal density satisfies a time-fractional diffusion equation.  For more information see mura-mainardi-generalized-grey-bm grey-brownian-motions.  The general method of solving non-linear Fokker-Planck equation via a solution of the Markovian Fokker-Planck was known to Sokolov sokolov-solving-nonmarkovian-fokker-planck-2002.  If $F(x,t)$ is a solution of $d/dt F(x,t) = A F(x,t)$ and we are interested in solving $d/dt P(x,t) = \int_0^t K(t-s) AF(x,s) ds$ then we take the Green’s function of the Markovian solution and find $P(x,t) = \int_0^\infty G(x,\tau) T(\tau,t) d\tau$ where

$LT(\tau,u) = \frac{1}{LK(u)}\exp(-\tau \frac{u}{LK(u)}$

In general the inversion of this Laplace transform is difficult.  Srokowski eliminates $T(\tau,t)$ by applying a Fourier transform.

## SROKOWSKI 2006 LEVY FRACTIONAL DIFFUSION SOLUTION

### Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

5:44 PM (8 minutes ago)

 to David, bcc: aimee
Srokowski considers a relatively complicated non-Markovian Levy Fokker-Planck equation and finds the Fourier transform of its solution

He considers the non-Markov equation where D^{gamma-1}_t refers to Riemann-Liouville derivative
d/dt p(x,t) = D^{gamma-1}_t d^mu/d|x|^mu ( |x|^{-theta} p(x,t) )  (*)
letting alpha = mu/(mu+theta) finds that the Fourier transform of p(t,x), say Fp(t,xi) has the exact solution

Fp( t, xi ) = 1 – (a0 * gamma^alpha * Gamma(1+alpha))/(Gamma(2*alpha+gamma*alpha-1)) * |xi|^mu * t^{2*alpha-gamma*alpha}

Now (*) is our recommendation for the linear fractional Levy diffusion without his theta term which is a position-dependent diffusion coefficient unnecessary for us.  So we can use his solution directly with his alpha set to 1, i.e.
we can use

Fp(t,xi) = 1 – a0*gamma*Gamma(2)/Gamma(1+gamma) * |xi|^mu * t^(2-gamma)

as the solution of the Levy distributed case which is the direct implementation of a non-Markovian Fokker-Planck equation with Levy stable density and time fractionation.  Now the caveat is that one has to worry about slow versus fast diffusions, which is a case that Mura et. al. handle and for which the gray Brownian motion densities handle.  Regardless, this Srokowski solution for power law memory kernel is fast and easy to fit and estimate its parameters on actual data with infrastructure I’ve built already —

def srokowski( ci, mu, gam):
return (1.0 – a0*gam*gamma(2.)/gamma(1+gam)*abs(xi)**mu*t**(2-gam))
The attached code implements this solution.  This is now a serious model for continuous time finance.   Here’s a series of 500 return point’s in maximum likelihood estimation.  So this is essentially the basic non-Markovian model for finance following Mandelbrot 1963 rather than fractional Brownian motion.  The theoretical Hurst exponent for memory kernel K(t) = t^{beta-1}/Gamma(beta) is simply H=beta/2.
So conceptually this seems reasonable.  The maximum likelihood estimation requires more effort – as these minimum -logliklihood figures are not yet clear to me.

(‘parameters ‘, array([-1.67020143,  3.14432051]))
[-1.67020143  3.14432051]
466.829922136
(‘parameters ‘, array([-1.67056341,  3.14411386]))
[-1.67056341  3.14411386]
465.203836304
(‘parameters ‘, array([-1.67050356,  3.14398995]))
[-1.67050356  3.14398995]
466.746321604
(‘parameters ‘, array([-1.6704804 ,  3.14418101]))
[-1.6704804   3.14418101]
465.261016721
(‘parameters ‘, array([-1.67049584,  3.14405364]))
3 Attachments