Contents; Abbreviations; Introduction; 1 Random variables and probability distributions; 1.1 Particle descriptions of partial differential equations; 1.2 Random variables and stochastic processes; 1.3 The n-point probability distributions; 1.4 Simple averages and scaling; 1.5 Pair correlations...

Machine generated contents note: 1. Random variables and probability distributions; 2. Martingales, Markov, and nonstationarity; 3. Stochastic calculus; 4. Ito processes and Fokker-Planck equations; 5. Selfsimilar Ito processes; 6. Fractional Brownian motion; 7. Kolmogorov's PDEs and...

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...

We show that our generalization of the Black-Scholes partial differential equation (pde) for nontrivial diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, this was proven for the case of the Gaussian logarithmic returns model by Harrison...

The distribution of price returns for a class of uncorrelated diffusive dynamics is considered. The basic assumptions are (1) that there is a "consensus" value associated with a stock, and (2) that the rate of diffusion depends on the deviation of the stock price from the consensus value. We...