?
Statistical Estimation of the Jump Activity for Time-changed Levy Processes
.
This paper is devoted to studying the problem of the statistical inference on the activity of jumps for a class of the so-called time-changed Levy processes, i.e., for the processes in the form Ys = XT (s), where X is a Levy process and T is a non-negative and non-decreasing stochastic process, which is referred to as time change. First, starting from some natural assumptions on the Levy measure of X, we infer on the asymptotic behavior of the characteristic function of Y. Next, we present a new method, which allows to consistently estimate the activity of small jumps in the dicult case of lowfrequency data.
In book
М.: ИППИ РАН, 2012.
Morozova E., Panov V., Finance Research Letters 2025 Vol. 86 No. A Article 108301
This paper presents a new approach to modeling the Bitcoin prices using the Lévy processes - a class of stochastic processes that are able to realistically capture the jump-type dynamics of financial time series. Our method is inspired by recent research on Bitcoin, which suggests that the prices are closely connected to the media attention ...
Added: September 4, 2025
Morozova E., Panov V., / Series SSRN "ERN: Speculation in Economic Markets". 2025. No. 5222389.
This paper presents a new approach to modeling the Bitcoin prices using the Lévy processes - a class of stochastic processes that are able to realistically capture the jump-type dynamics of financial time series. Our method is inspired by the findings in recent research on Bitcoin, which suggest that the prices are closely connected to ...
Added: May 2, 2025
Morozova E., Panov V., Applied Stochastic Models in Business and Industry 2023 Vol. 39 No. 6 P. 772–788
In this paper, we present a new bivariate model for the joint description of the Bitcoin prices and the media attention to Bitcoin. Our model is based on the class of the Levy processes and is able to realistically reproduce the jump-type dynamics of the considered time series. We focus on the lowfrequency setup, which ...
Added: June 29, 2023
Morozova E., Panov V., / Series arXiv.org "q-fin.ST". 2022. No. 2210.13824.
In this paper, we present a new bivariate model for the joint description of the Bitcoin prices and the media attention to Bitcoin. Our model is based on the class of the Lévy processes and is able to realistically reproduce the jump-type dynamics of the considered time series. We focus on the low-frequency setup, which ...
Added: October 26, 2022
Lubashevsky I., Heuer A., Friedrich R. et al., The European Physical Journal B 2010 Vol. 78 P. 207–216
We consider a previously devised model describing Lévy random walks [I. Lubashevsky, R. Friedrich, A. Heuer, Phys. Rev. E 79, 011110 (2009); I. Lubashevsky, R. Friedrich, A. Heuer, Phys. Rev. E 80, 031148 (2009)]. It is demonstrated numerically that the given model describes Lévy random walks with superdiffusive, ballistic, as well as superballistic dynamics. Previously ...
Added: November 6, 2021
Lubashevsky I., The European Physical Journal B 2010 Vol. 82 P. 189–195
A continuous Markovian model for truncated Lévy flights is proposed. It generalizes the approach developed previously by Lubashevsky et al. [Phys. Rev. E 79, 011110 (2009); Phys. Rev. E 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010)] and allows for nonlinear friction in wandering particle motion as well as saturation of the noise intensity ...
Added: November 6, 2021
Lubashevsky I., Physica A: Statistical Mechanics and its Applications 2013 Vol. 392 No. 10 P. 2323–2346
The paper is devoted to the relationship between the continuous Markovian description of Lévy flights developed previously (see, e.g., I.A. Lubashevsky, Truncated Lévy flights and generalized Cauchy processes, Eur. Phys. J. B 82 (2011) 189–195 and references therein) and their equivalent representation in terms of discrete steps of a wandering particle, a certain generalization of ...
Added: November 6, 2021
Lubashevsky I., Friedrich R., Heuer A., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2009 Vol. 80 Article 031148
The paper presents a multidimensional model for nonlinear Markovian random walks that generalizes the one we developed previously [I. Lubashevsky, R. Friedrich, and A. Heuer, Phys. Rev. E 79, 011110 (2009)] in order to describe the Lévy-type stochastic processes in terms of continuous trajectories of walker motion. This approach may open a way to treat ...
Added: November 6, 2021
Lubashevsky I., Friedrich R., Heuer A., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2009 Vol. 79 Article 011110
Based on multivariate Langevin processes we present a realization of Lévy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity-dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Lévy flights. Our procedure is similar to ...
Added: November 6, 2021
Finkelstein D., Kondratiev Y., Molchanov S. et al., Stochastics and Dynamics 2018 Vol. 18 No. 05 P. 1850037
We study stability of stationary solutions for a class of nonlocal semilinear parabolic equations. To this end, we prove the Feynman–Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation ...
Added: November 14, 2019
Manita A., Journal of Physics: Conference Series 2019 Vol. 1163 No. 012060 P. 1–7
L´evy stochastic processes and related fine analytic properties of probability distributions such as infinite divisibility play an important role in construction of stochastic models of various distributed networks (e.g., local clock synchronization), of some physical systems (e.g., anomalous diffusions, quantum probability models), of finance etc. Nevertheless, little is known about limit probability laws resulted from ...
Added: June 21, 2019
Belomestny D., GUGUSHVILI S., SCHAUER M. et al., Communications in Mathematical Sciences 2019 Vol. 17 No. 3 P. 781–816
Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Levy density of a Levy process belonging to a flexible class of infinite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that ...
Added: June 11, 2019
Kelbert M., Moreno-Franco H. A., SIAM Journal on Control and Optimization 2019 Vol. 57 No. 3 P. 2185–2213
In this paper, we guarantee the existence and uniqueness (in the almost everywhere
sense) of the solution to a Hamilton-Jacobi-Bellman (HJB) equation with gradient
constraint and a partial integro-di erential operator whose Levy measure has bounded
variation. This type of equation arises in a singular control problem, where the state
process is a multidimensional jump-di usion with jumps of ...
Added: February 13, 2019
Gushchin A. A., Kordzakhia N., Novikov A., Statistical Inference for Stochastic Processes 2018 Vol. 21 No. 2 P. 363–383
We provide a full description of the class of translation invariant experiments with independent increments. Necessary and sufficient conditions for the weak convergence and the comparison of experiments within this class are given. Finally, we prove exponential boundedness of Pitman estimators in these models. ...
Added: June 29, 2018
Belomestny D., Trabs M., Annales de l'institut Henri Poincare (B) Probability and Statistics 2018 Vol. 54 No. 3 P. 1583–1621
The estimation of the diffusion matrix Σ of a high-dimensional, possibly time-changed Levy process is studied, based on discrete observations of the process with a fixed distance. A low-rank condition is imposed on Σ. Applying a spectral approach, we construct a weighted least-squares estimator with nuclear-norm-penalisation. We prove oracle inequalities and derive convergence rates for ...
Added: May 5, 2018
Belomestny D., Orlova T., Panov V., Statistica Neerlandica 2019 Vol. 1 P. 100–117
We consider a new method of semiparametric statistical
estimation for the continuous-time moving-average Lévy
processes. We derive the convergence rates of the proposed
estimators and show that these rates are optimal in
minimax sense. ...
Added: April 28, 2018
Kelbert M., Karpikov I., Science and Business: Ways of Development 2018 Vol. 79 No. 1 P. 56–68
This article gives a brief summary on the main theoretical and practical results for the Scale functions. The article is organized in the following way: the first part describes the main theoretical concepts of Lévy processes, gives the formal definition and analytical properties of the Scale function. The second part describes the most significant practical ...
Added: April 5, 2018
Belomestny D., Panov V., Woerner J., Bernoulli: a journal of mathematical statistics and probability 2019 Vol. 25 No. 2 P. 902–931
In this paper we study the problem of statistical inference for a continuous-time moving average L\'evy process of the form
\[ Z_{t}=\int_{\R}\mathcal{K}(t-s)\, dL_{s},\quad t\in\mathbb{R}, \] with a deterministic kernel \(\K\) and a L{\'e}vy process \(L\). Especially the estimation of the L\'evy measure \(\nu\) of $L$ from low-frequency observations of the process $Z$ is considered. We construct ...
Added: December 9, 2017
Manita A., , in: New trends in Stochastic Modeling and Data Analysis.: Athens: ISAST: International Society for the Advancement of Science and Technology, 2015.
Added: June 20, 2017
Belomestny D., Orlova T., Panov V., / Series arXiv "stat". 2017. No. 1702.02794.
We consider a new method of the semiparametric statistical estimation for the continuous-time moving average L\'evy processes. We derive the convergence rates of the proposed estimators, and show that these rates are optimal in the minimax sense. ...
Added: February 10, 2017