We analyze intraday fluctuations in several stock indices to investigate the underlying stochastic processes using techniques appropriate for processes with nonstationary increments. The five most actively traded stocks each contains two time intervals during the day where the variance of...

Recent reports suggest that the stochastic process underlying financial time series is nonstationary with nonstationary increments. Therefore, time averaging techniques through sliding intervals are inappropriate and ensemble methods have been proposed. Using daily ensemble averages we analyze...

Increments in financial markets have anomalous statistical properties including fat-tailed distributions and volatility clustering (i.e., the autocorrelation functions of return increments decay quickly but those of the squared increments decay slowly). One of the central questions in financial...

Arguably the most important problem in quantitative finance is to understand the nature of stochastic processes that underlie market dynamics. One aspect of the solution to this problem involves determining characteristics of the distribution of fluctuations in returns. Empirical studies...

We show that our earlier 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, the equivalence of Black-Scholes to a Martingale was proven for the case of...

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. We construct a large set of scaling solutions of Fokker-Planck partial differential equations where H≠1/2. Thus Markov...

There is much confusion in the literature over Hurst exponents. Recently, we took a step in the direction of eliminating some of the confusion. One purpose of this paper is to illustrate the difference between fBm on the one hand and Gaussian Markov processes where H≠1/2 on the other. The...