Analyzing the Fine Structure of Continuous Time Stochastic Processes
In the recent years especially in finance many different models either based on semimartingales, purely continuous, pure jump and a mixture of both, or fractional Brownian motion have been proposed in the literature. We provide a class of easily computable estimators which allows to infer the fine structure of the underlying process in terms of the Blumenthal-Getoor index or the Hurst exponent based on high frequency data. This method makes it possible not only to detect jumps, but also determine their activity and the regularity of continuous components, which can be used for model selection or to analyze the market microstructure by taking into account different time scales. Furthermore, our method provides a simple graphical tool for detecting jumps.
KeywordsStatistical inference Blumenthal-Getoor index Hurst exponent Lévy process fractional Brownian motion stochastic volatility model power variation high frequency data
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