We consider the problem of estimating the fractional order of a Lévy process from low frequency historical and options data. An estimation methodology is developed which allows us to treat both estimation and calibration problems in a unified way. The corresponding procedure consists of two...

We define a new concept termed the activity signature function, which is constructed from discrete observations of a process evolving continuously in time. Under quite general regularity conditions, we derive the asymptotic properties of the function as the sampling frequency increases and show...

The paper undertakes a non-parametric analysis of the very high frequency movements in stock market volatility using very finely sampled data on the S&P VIX index compiled by the CBOE. The data suggest that stock market volatility is best described as a pure jump process without a continuous...

We consider the problem of estimating the fractional order of a L´evy process from low frequency historical and options data. An estimation methodology is developed which allows us to treat both estimation and calibration problems in a unified way. The corresponding procedure consists of two...

This paper derives the asymptotic behavior of realized power variation of pure-jump It^o semimartingales as the sampling frequency within a fixed interval increases to infinity. We prove convergence in probability and an associated central limit theorem for the realized power variation as a...

Central place systems have been demonstrated to possess self-similarity in both the theoretical and empirical perspectives. A central place model can be treated as a monofractal with a single scaling process. However, a real system of human settlements is a complex network with multi-scaling...

The attractor in usual box-counting methods is partitioned with a fixed grid. We describe an algorithm whose boxes can move into the phase space and adapt to the geometry of the attractor. Lyapunov exponents and the singularity spectrum from time series are estimated using this algorithm.