Abstract
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 p-variation, 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.
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Norvaiša, R., Salopek, D.M. Estimating the p-Variation Index of a Sample Function: An Application to Financial Data Set. Methodology and Computing in Applied Probability 4, 27–53 (2002). https://doi.org/10.1023/A:1015753313674
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DOI: https://doi.org/10.1023/A:1015753313674