Linear vs. Nonlinear Diffusion and Martingale Option Pricing

First, classes of Markov processes that scale exactly with a Hurst exponent H are derived in closed form. A special case of one class is the Tsallis density, advertised elsewhere as nonlinear diffusion or diffusion with nonlinear feedback. But the Tsallis model is only one of a very large class of linear diffusion with a student-t like density. Second, we show by stochastic calculus that our generalization of the Black-Scholes partial differential equation (pde) for variable diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, this was proven for the restricted case of Gaussian logarithmic returns by Harrison and Kreps, but we prove it here for large classes of empirically useful and theoretically interesting returns models where the diffusion coefficient D(x,t) depends on both logarithmic returns x and time t. Finally, we prove that option prices blow up if fat tails in returns x are included in the market distribution.