Measuring the self-similarity exponent in Lévy stable processes of financial time series

2013 ◽  
Vol 392 (21) ◽  
pp. 5330-5345 ◽  
Author(s):  
M. Fernández-Martínez ◽  
M.A. Sánchez-Granero ◽  
J.E. Trinidad Segovia
Keyword(s):  
Time Series ◽  
Financial Time Series ◽  
The Self ◽  
Stable Processes ◽  
Self Similarity ◽  
Financial Time

scholarly journals Fractional Laplace motion

2006 ◽  
Vol 38 (02) ◽  
pp. 451-464 ◽  
Author(s):  
T. J. Kozubowski ◽  
M. M. Meerschaert ◽  
K. Podgórski
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Hydraulic Conductivity ◽  
Fractional Brownian Motion ◽  
Long Range ◽  
Financial Time Series ◽  
Long Range Dependence ◽  
Self Similarity ◽  
One Dimensional ◽  
Financial Time

Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

scholarly journals Fractional Laplace motion

2006 ◽  
Vol 38 (2) ◽  
pp. 451-464 ◽  
Author(s):  
T. J. Kozubowski ◽  
M. M. Meerschaert ◽  
K. Podgórski
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Hydraulic Conductivity ◽  
Fractional Brownian Motion ◽  
Long Range ◽  
Financial Time Series ◽  
Long Range Dependence ◽  
Self Similarity ◽  
One Dimensional ◽  
Financial Time

Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

scholarly journals FRACTAL GEOMETRY OF FINANCIAL TIME SERIES

Fractals ◽  
1995 ◽  
Vol 03 (03) ◽  
pp. 609-616 ◽  
Author(s):  
CARL J.G. EVERTSZ
Keyword(s):  
Time Series ◽  
Stock Price ◽  
Financial Time Series ◽  
Quantitative Measure ◽  
Scaling Behavior ◽  
The Self ◽  
Self Similarity ◽  
Absolute Size ◽  
Stable Random Variables ◽  
German Stock

A simple quantitative measure of the self-similarity in time-series in general and in the stock market in particular is the scaling behavior of the absolute size of the jumps across lags of size k. A stronger form of self-similarity entails that not only this mean absolute value, but also the full distributions of lag-k jumps have a scaling behavior characterized by the above Hurst exponent. In 1963, Benoit Mandelbrot showed that cotton prices have such a strong form of (distributional) self-similarity, and for the first time introduced Lévy’s stable random variables in the modeling of price records. This paper discusses the analysis of the self-similarity of high-frequency DEM-USD exchange rate records and the 30 main German stock price records. Distributional self-similarity is found in both cases and some of its consequences are discussed.

A WAVELET METHOD COUPLED WITH QUASI-SELF-SIMILAR STOCHASTIC PROCESSES FOR TIME SERIES APPROXIMATION

2011 ◽  
Vol 09 (05) ◽  
pp. 685-711 ◽  
Author(s):  
MOHAMED ESSAIED HAMRITA ◽  
NIDHAL BEN ABDALLAH ◽  
ANOUAR BEN MABROUK
Keyword(s):  
Time Series ◽  
Stochastic Processes ◽  
Scaling Laws ◽  
Financial Time Series ◽  
Wavelet Theory ◽  
Self Similarity ◽  
Wavelet Method ◽  
Financial Time ◽  
Functional Methods ◽  
Self Similar

Scaling laws and generally self-similar structures are now well known facts in financial time series. Furthermore, these signals are characterized by the presence of stochastic behavior allowing their analysis with pure functional methods being incomplete. In the present paper, some existing models are reviewed and modified, based on wavelet theory and self-similarity, to recover multi-scaling cases for approximating financial signals. The resulting models are then tested on some empirical examples and analyzed for error estimates.

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