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

Abstract

We show by explicit closed form calculations that a Hurst exponent H ≠ 1 2 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 H ≠ 1 2 . Thus Markov processes, which by construction have no long time correlations, can have H ≠ 1 2 . If a Markov process scales with Hurst exponent H ≠ 1 2 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 H ≠ 1 2 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 H ≠ 1 2 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.