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

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1576
Author(s):  
Jarosław Klamut ◽  
Tomasz Gubiec
Keyword(s):  
Random Walk ◽  
Stock Market ◽  
Autocorrelation Function ◽  
Continuous Time ◽  
Crucial Role ◽  
Continuous Time Random Walk ◽  
Time Intervals ◽  
Stock Market Returns ◽  
Volatility Clustering

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.

scholarly journals Turbulent pair dispersion as a continuous-time random walk

2014 ◽  
Vol 755 ◽  
Author(s):  
Simon Thalabard ◽  
Giorgio Krstulovic ◽  
Jérémie Bec
Keyword(s):  
Random Walk ◽  
High Resolution ◽  
Probability Distribution ◽  
Continuous Time ◽  
Eddy Diffusivity ◽  
Continuous Time Random Walk ◽  
Relative Dispersion ◽  
Time Intervals

AbstractThe phenomenology of turbulent relative dispersion is revisited. A heuristic scenario is proposed, in which pairs of tracers undergo a succession of independent ballistic separations during time intervals whose lengths fluctuate. This approach suggests that the logarithm of the distance between tracers self-averages and performs a continuous-time random walk. This leads to specific predictions for the probability distribution of separations, which differ from those obtained using scale-dependent eddy-diffusivity models (e.g. in the framework of Richardson’s approach). These predictions are tested against high-resolution simulations and shed new light on the explosive separation between tracers.

Continuous Time Random Walk and Migration–Proliferation Dichotomy of Brain Cancer

2015 ◽  
Vol 10 (01) ◽  
pp. 37-57 ◽  
Author(s):  
A. Iomin
Keyword(s):  
Random Walk ◽  
Brain Cancer ◽  
Continuous Time ◽  
Continuous Time Random Walk ◽  
Suggested Model ◽  
Cell Dynamics ◽  
Fractional Kinetics ◽  
And Migration ◽  
Kinetics Of

A theory of fractional kinetics of glial cancer cells is presented. A role of the migration-proliferation dichotomy in the fractional cancer cell dynamics in the outer-invasive zone is discussed and explained in the framework of a continuous time random walk. The main suggested model is based on a construction of a 3D comb model, where the migration-proliferation dichotomy becomes naturally apparent and the outer-invasive zone of glioma cancer is considered as a fractal composite with a fractal dimension Dfr< 3.

Correlated continuous-time random walk in the velocity field: the role of velocity and weak asymptotics

Soft Matter ◽  
2021 ◽  
Author(s):  
Jian Liu ◽  
Caiyun Zhang ◽  
Jing-Dong Bao ◽  
Xiaosong Chen
Keyword(s):  
Random Walk ◽  
Velocity Field ◽  
Waiting Time ◽  
Continuous Time ◽  
Fluid Velocity ◽  
Time Correlation ◽  
Space Time ◽  
Continuous Time Random Walk ◽  
Weak Asymptotics

Within the framework of space-time correlated continuous-time random walk model, anomalous diffusion of particle moving in the velocity field is studied in this paper. The weak asymptotic form ω(t) ∼ t−(1+α); 1 < α < 2 for large t, is considered to be the waiting time distribution. Analytical results reveal that the diffusion in the velocity field, i.e., the mean squared displacement, can display a multi-fractional form caused by dispersive bias and space-time correlation. Numerical results indicate that the multi-fractional diffusion leads to a crossover phenomenon in-between the process at intermediate timescale, followed by a steady state which is always determined by the largest diffusion exponent term. In addition, the role of velocity and weak asymptotics is discussed. The extremely small fluid velocity can make the diffusion to be characterized by diffusion coefficient instead of diffusion exponent, which is distinctly different from the former definition. Especially, for the waiting time displaying weak asymptotic property, if the anomalous part is suppressed by the normal part, a second crossover phenomenon appears at intermediate timescale, followed by a steady normal diffusion, which implies that the anomalies underlying the process are smoothed out at large timescale. Moreover, we discuss that the consideration of bias and correlation could help to avoid a possible not readily noticeable mistake in studying the topic concerned in this paper, which may be helpful for the relevant experimental research.

Quartile histogram assessment of glioma malignancy using high b ‐value diffusion MRI with a continuous‐time random‐walk model

2021 ◽  
Vol 34 (4) ◽  
Author(s):  
M. Muge Karaman ◽  
Jiaxuan Zhang ◽  
Karen L. Xie ◽  
Wenzhen Zhu ◽  
Xiaohong Joe Zhou
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Diffusion Mri ◽  
B Value ◽  
Continuous Time Random Walk ◽  
Random Walk Model ◽  
High B Value

Multiscale Modeling of Carbonate Acidizing Using Continuous Time Random Walk Approach

2017 ◽  
Author(s):  
Kang Kang ◽  
Elsayed Abdelfatah ◽  
Maysam Pournik ◽  
Bor Jier Shiau ◽  
Jeffrey Harwell
Keyword(s):  
Random Walk ◽  
Multiscale Modeling ◽  
Continuous Time ◽  
Continuous Time Random Walk ◽  
Carbonate Acidizing

A note on passage times and infinitely divisible distributions

1967 ◽  
Vol 4 (2) ◽  
pp. 402-405 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Time Intervals ◽  
Mutually Independent ◽  
Passage Times

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

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