Variable Step Random Walks and Self-Similar Distributions

Gunaratne, Gemunu H.; McCauley, Joseph L.; Nicol, Matthew; Török, Andrei

doi:10.1007/s10955-005-5474-y

Cited by 12 publications

(16 citation statements)

References 22 publications

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“…1. When H ϭ 1 ⁄2, the diffusion coefficient has been shown to be a function of u; i.e., (12,13). When H 1 ⁄2, we can ''rescale'' time intervals by ˜ϭ 2 H (8,14).…”

Section: Resultsmentioning

confidence: 99%

“…Its distribution has a scaling index H and a scaling function F(u) ϭ 1 ⁄2 exp (Ϫ͉u͉) (12,13). (See the discussion after Eq.…”

Section: Resultsmentioning

confidence: 99%

“…The stochastic dynamics within these scaling intervals was shown to be diffusive with a diffusion coefficient that depends on both time and the exchange rate (2,12). We presented a detailed analysis of one of the scaling regions that begins at 9:00 a.m. New York time and last for Ϸ3 h. The dynamical scaling index for the variable diffusion process here was shown to be Ϸ0.35, significantly lower than the sliding interval value H S Ϸ 0.5 reported in previous analyses of financial markets.…”

Section: Discussionmentioning

confidence: 99%

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Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets

Bassler

McCauley

Gunaratne

2007

Proc. Natl. Acad. Sci. U.S.A.

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139

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Fat-tailed distributions have been reported in fluctuations of financial markets for more than a decade. Sliding interval techniques used in these studies implicitly assume that the underlying stochastic process has stationary increments. Through an analysis of intraday increments, we explicitly show that this assumption is invalid for the Euro-Dollar exchange rate. We find several time intervals during the day where the standard deviation of increments exhibits power law behavior in time. Stochastic dynamics during these intervals is shown to be given by diffusion processes with a diffusion coefficient that depends on time and the exchange rate. We introduce methods to evaluate the dynamical scaling index and the scaling function empirically. In general, the scaling index is significantly smaller than previously reported values close to 0.5. We show how the latter as well as apparent fat-tailed distributions can occur only as artifacts of the sliding interval analysis.fat tails ͉ Fokker-Planck equation ͉ Langevin simulations A rguably the most important problem in quantitative finance is to understand the nature of stochastic processes that underlie market dynamics. One aspect of the solution to this problem involves determining characteristics of the distribution of fluctuations in returns. Empirical studies conducted over the last decade have reported that they are non-Gaussian, scale in time, and have power law (or fat) tails (1-6). However, by combining increments at multiple times in their statistical analyses (sliding interval techniques), these studies implicitly assume that the stochastic process has stationary increments. For financial markets, it is not clear whether this assumption is valid. For example, it is possible that trading activity at the beginning of a trading day may differ from that at the end of the day. How is it possible to test whether intraday fluctuations are timeindependent? If they are time-dependent, how can statistical analyses be conducted? Will results from previous studies be invalidated?Our analysis is conducted on intraday Euro-Dollar exchange rates (traded 24 h per day) during 1999-2004 recorded in 1-min intervals. It is based on the assumption that intraday variations in the market follow the same underlying stochastic process every day. Then, a statistical analysis for fluctuations at a given time of the day can be conducted by using data from multiple trading days within the sample.We find from this analysis the following. (i) The stochastic process is time-dependent and there are several intervals during the day where the standard deviation of increments exhibits power law behavior. Stochastic dynamics during these intervals is given by variable diffusion processes (2). (ii) Dynamical scaling indices and empirical scaling functions within these scaling intervals are different from previously reported results. We show how the latter can result from the application of sliding interval methods to a time-dependent stochastic process. (iii) Autocorrelation functions for var...

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“…1. When H ϭ 1 ⁄2, the diffusion coefficient has been shown to be a function of u; i.e., (12,13). When H 1 ⁄2, we can ''rescale'' time intervals by ˜ϭ 2 H (8,14).…”

Section: Resultsmentioning

confidence: 99%

“…Its distribution has a scaling index H and a scaling function F(u) ϭ 1 ⁄2 exp (Ϫ͉u͉) (12,13). (See the discussion after Eq.…”

Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets

Bassler

McCauley

Gunaratne

2007

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

139

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“…They account for finite-time corrections to the scaling behavior, and will be applied to standard and scaled versions of example stochastic processes to highlight the roles played by H, L, J, and M . Finally, exponents for a model of intraday trading in financial markets, variable diffusion processes [23,[34][35][36][37][38][39][40][41], will be computed and compared to the results with an empirical analysis of financial market data.…”

mentioning

confidence: 99%

Anomalous scaling of stochastic processes and the Moses effect

et al. 2017

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The state of a stochastic process evolving over a time t is typically assumed to lie on a normal distribution whose width scales like t 1 2 . However, processes where the probability distribution is not normal and the scaling exponent differs from 1 2 are known. The search for possible origins of such "anomalous" scaling and approaches to quantify them are the motivations for the work reported here. In processes with stationary increments, where the stochastic process is time-independent, auto-correlations between increments and infinite variance of increments can cause anomalous scaling. These sources have been referred to as the Joseph effect the Noah effect, respectively. If the increments are non-stationary, then scaling of increments with t can also lead to anomalous scaling, a mechanism we refer to as the Moses effect. Scaling exponents quantifying the three effects are defined and related to the Hurst exponent that characterizes the overall scaling of the stochastic process. Methods of time series analysis that enable accurate independent measurement of each exponent are presented. Simple stochastic processes are used to illustrate each effect. Intraday financial time series data is analyzed, revealing that its anomalous scaling is due only to the Moses effect. In the context of financial market data, we reiterate that the Joseph exponent, not the Hurst exponent, is the appropriate measure to test the efficient market hypothesis.

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“…Instead, it is a Markovian effect with a non-trivial variable diffusion coefficient [22], or more precisely, an |X t | dependent diffusion coefficient D(|X t |, ·) [25,26]. This phenomenon rules out statistically independent processes including Lévy processes [26]. It also rules out other models, e.g., stochastic volatility model [27].…”

Section: Evidence For the Variable Diffusion Modelmentioning

confidence: 99%