Markov Processes, Hurst Exponents, and Nonlinear Diffusion Equations with application to finance

摘要

We show by explicit closed form calculations that a Hurst exponent H that isnot 1/2 does not necessarily imply long time correlations like those found infractional Brownian motion. We construct a large set of scaling solutions ofFokker-Planck partial differential equations where H is not 1/2. Thus Markovprocesses, which by construction have no long time correlations, can have H notequal to 1/2. If a Markov process scales with Hurst exponent H then it simplymeans that the process has nonstationary increments. For the scaling solutions,we show how to reduce the calculation of the probability density to a singleintegration once the diffusion coefficient D(x,t) is specified. As an example,we generate a class of student-t-like densities from the class of quadraticdiffusion coefficients. Notably, the Tsallis density is one member of thatlarge class. The Tsallis density is usually thought to result from a nonlineardiffusion equation, but instead we explicitly show that it follows from aMarkov process generated by a linear Fokker-Planck equation, and therefore froma corresponding Langevin equation. Having a Tsallis density with H not equal to1/2 therefore does not imply dynamics with correlated signals, e.g., like thoseof fractional Brownian motion. A short review of the requirements forfractional Brownian motion is given for clarity, and we explain why the usualsimple argument that H unequal to 1/2 implies correlations fails for Markovprocesses with scaling solutions. Finally, we discuss the question of scalingof the full Green function g(x,t;x',t') of the Fokker-Planck pde.