Markov processes, Hurst exponents, and nonlinear diffusion equations: With application to finance

We show by explicit closed form calculations that a Hurst exponent does not necessarily imply long time correlations like those found in fractional Brownian motion (fBm). We construct a large set of scaling solutions of Fokker–Planck partial differential equations (pdes) where . Thus Markov processes, which by construction have no long time correlations, can have . If a Markov process scales with Hurst exponent then it simply means that the process has nonstationary increments. For the scaling solutions, we show how to reduce the calculation of the probability density to a single integration 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 quadratic diffusion coefficients. Notably, the Tsallis density is one member of that large class. The Tsallis density is usually thought to result from a nonlinear diffusion equation, but instead we explicitly show that it follows from a Markov process generated by a linear Fokker–Planck equation, and therefore from a corresponding Langevin equation. Having a Tsallis density with therefore does not imply dynamics with correlated signals, e.g., like those of fBm. A short review of the requirements for fBm is given for clarity, and we explain why the usual simple argument that implies correlations fails for Markov processes with scaling solutions. Finally, we discuss the question of scaling of the full Green function g(x,t;x′,t′) of the Fokker–Planck pdes.