Estimating the p-variation index of a sample function: An application to financial data set

In this paper we apply a real analysis approach to test continuous time stochastic models of financial mathematics. Specifically, fractal dimension estimation methods are applied to statistical analysis of continuous time stochastic processes. To estimate a roughness of a sample function we modify a box-counting method typically used in estimating fractal dimension of a graph of a function. Here the roughness of a function f is defined as the infimum of numbers p > 0 such that f has bounded pvariation, which we call the p-variation index of f . The method is also tested on estimating the exponent α ∈ [1, 2] of a simulated symmetric α-stable process, and on estimating the Hurst exponent H ∈ (0, 1) of a simulated fractional Brownian motion.