Directed continuous-time random walk with memory

Klamut, Jarosław; Gubiec, Tomasz

doi:10.1140/epjb/e2019-90453-y

Cited by 3 publications

(4 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To derive the nonlinear ACF of absolute returns, we define the new CTRW process, and by calculating its linear ACF, we obtain the nonlinear ACF of price increments. Following [ 60 ], if as

we use only the positive half of the previous distribution multiplied by 2, we deal with the case of non-zero drift and obtain an artificial, monotonically increasing process. As

, we obtain the slow power-law decay of the autocorrelation of absolute returns, as in the empirical results presented as a solid black line in Figure 2 b.…”

Section: Resultsmentioning

confidence: 99%

“…variables and only the dependence between

and

is considered. Unfortunately, models considering only this type of dependencies turned out to be unable to describe the time ACF of absolute values of price changes [ 60 ]. Technically, it is possible to obtain a CTRW model reproducing both stylized facts, but it requires a power-law waiting-time distribution

.…”

Section: Motivationmentioning

confidence: 99%

“…The CTRW models with correlated increments were initially proposed to study lattice gases [ 45 , 46 , 47 ]. More recently, they have been used to model high-frequency financial data [ 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 ]. On the other hand, CTRW models with correlated waiting times are not well-studied.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Continuous Time Random Walk with Correlated Waiting Times. The Crucial Role of Inter-Trade Times in Volatility Clustering

Klamut

Gubiec

2021

Entropy

Self Cite

View full text Add to dashboard Cite

In many physical, social, and economic phenomena, we observe changes in a studied quantity only in discrete, irregularly distributed points in time. The stochastic process usually applied to describe this kind of variable is the continuous-time random walk (CTRW). Despite the popularity of these types of stochastic processes and strong empirical motivation, models with a long-term memory within the sequence of time intervals between observations are rare in the physics literature. Here, we fill this gap by introducing a new family of CTRWs. The memory is introduced to the model by assuming that many consecutive time intervals can be the same. Surprisingly, in this process we can observe a slowly decaying nonlinear autocorrelation function without a fat-tailed distribution of time intervals. Our model, applied to high-frequency stock market data, can successfully describe the slope of decay of the nonlinear autocorrelation function of stock market returns. We achieve this result without imposing any dependence between consecutive price changes. This proves the crucial role of inter-event times in the volatility clustering phenomenon observed in all stock markets.

show abstract

“…To derive the nonlinear ACF of absolute returns, we define the new CTRW process, and by calculating its linear ACF, we obtain the nonlinear ACF of price increments. Following [ 60 ], if as

we use only the positive half of the previous distribution multiplied by 2, we deal with the case of non-zero drift and obtain an artificial, monotonically increasing process. As

, we obtain the slow power-law decay of the autocorrelation of absolute returns, as in the empirical results presented as a solid black line in Figure 2 b.…”

Section: Resultsmentioning

confidence: 99%

“…variables and only the dependence between

and

.…”

Section: Motivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Continuous Time Random Walk with Correlated Waiting Times. The Crucial Role of Inter-Trade Times in Volatility Clustering

Klamut

Gubiec

2021

Entropy

Self Cite

View full text Add to dashboard Cite

show abstract

“…It may be the result of the existence of long-range correlations between subsequent inter-event times (cf. [58] and references therein). These correlations, presumably, can be the source of real multifractality covered herein.…”

Section: Normalized Multifractal Detrended Fluctuation Analysismentioning

confidence: 99%

The new face of multifractality: Multi-branchedness and the phase transitions in time series of mean inter-event times

Klamut,

Kutner,

Gubiec

et al. 2018

Preprint

Self Cite

View full text Add to dashboard Cite

We develop an extended multifractal analysis based on the Legendre-Fenchel transform (sometimes referred to as Legendre multi-branched one) rather than the routinely used canonical Legendre transform. In our variant of coarse-graining pre-processing, the local detrending of time series has been replaced by an appropriate averaging over days combined with properly-suited detrending on a daily time scale. This new approach is devoid of troublesome artifacts in the form of innumerable faults of these local trends that can deform the hierarchy of fluctuations and hence the final multifractality. Notably, our analysis is sensitive to the change of time scale as it should be. This analysis has developed, e.g., for empirical time series of inter-event or waiting times, which are an essential element of the popular continuous-time random walk formalism. The core of this extended multifractal analysis is the non-monotonic behavior of the generalized Hurst exponent -the fundamental exponent of the study -and hence a multi-branched spectrum of dimensions, which for our case is additionally of the left-sided one. We examine the main thermodynamic consequences of the existence of this type of multifractality. They can be expressed directly in the language of thermally stable, metastable, and unstable phases, and phase transitions between them as well. These phase transitions are of the first and second orders according to the modified Ehrenfest classification, sometimes called the Mandelbrot one.

show abstract

Multibranch multifractality and the phase transitions in time series of mean interevent times

Klamut

Kutner²,

Gubiec

et al. 2020

Phys. Rev. E

View full text Add to dashboard Cite

Directed continuous-time random walk with memory

Cited by 3 publications

References 52 publications

Continuous Time Random Walk with Correlated Waiting Times. The Crucial Role of Inter-Trade Times in Volatility Clustering

Continuous Time Random Walk with Correlated Waiting Times. The Crucial Role of Inter-Trade Times in Volatility Clustering

The new face of multifractality: Multi-branchedness and the phase transitions in time series of mean inter-event times

Multibranch multifractality and the phase transitions in time series of mean interevent times

Contact Info

Product

Resources

About