• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site
  • HSE University
  • Research projects
  • Analysis of stochastic events described by the fractional stochastic equations, Langevin dynamics with elastic collisions, analysis of extremal events

Analysis of stochastic events described by the fractional stochastic equations, Langevin dynamics with elastic collisions, analysis of extremal events

Priority areas of development: mathematics
The project has been carried out as part of the HSE Program of Fundamental Studies.

Goal of research

Modeling of complex events with random environment evolution and development in their parametric statistical inference.


In the first part of the project, the probabilistic method of upper functions was employed as the main tool when considering anomalous diffusions. Martingale methods have also been involved, along with stability theory and real analysis used to examine the solutions. For research on the second part of the project, concerning the analysis of fractional equations, we required the use of methods of complex and functional analysis, as well as the theory of martingales. In the third part of the project, which addresses the asset price modeling issue, the methods of asymptotic statistics, optimization methods, and measure theory proved to be necessary in order to estimate the parameters. The methodology of the fourth part of the project related to the optimal vaccine distribution in the epidemic model, was based on the application of the methods of the theory of ordinary differential equations and numerical methods of optimal control.

Empirical base of research

Bloomberg, Bureau van Dijk, COMPUSTAT (Global), Thomson Reuters Eikon and other financial databases.

Results of research

Anomalous diffusions governed by a time-varying Ornstein-Uhlenbeck process have been examined with the help of upper functions. The obtained upper functions almost surely majorate the displacement process. We have performed the diffusion classification (normal diffusion, sub - or superdiffusion) which depends on the behavior of their corresponding upper functions.

We applied a probabilistic approach to the analysis of solutions of a class of fractional differential equations. The possibility of a path integral representation stable with respect to initial conditions and main parameters has been shown. A characterization of the solutions by means of operator-valued analytic functions, being exit times of monotone Markov processes, has been carried out.

We proposed a model based on stable processes and dependences determined by Levy copulas. We have developed a simulation method that quite well represents a real correlation between stock prices.

The problem of the optimal vaccine sharing in an epidemic model with two populated centers has been considered. We have studied dependence of optimal vaccine distribution on model parameters (available stock of the vaccine, migration rate, population, etc.). Upon analysis, some recommendations on the optimal vaccination rules have been developed

Level of implementation, recommendations on implementation or outcomes of the implementation of the results/ Recommendations

As far as recommendations, it is proposed to verify the conditions, as well as to perform reliable verification of the models on real data.


Kelbert M., Sazonov I., Bocharov G., Grebennikov D., Meyerhans A. Viral Infection Dynamics Model Based on a MarkovProcess with Time Delay between Cell Infection and Progeny Production // Mathematics. 2020. Vol. 8. No. 8 doi
Grabchak M., Kelbert M., Paris Q. On the occupancy problem for a regime switching model // Journal of Applied Probability. 2020. Vol. 57. No. 1. P. 53-77. doi
Veretennikov A. On mean-field GI/GI/1 queueing model: existence, uniqueness // Queueing Systems. 2020. Vol. 94. No. 3. P. 243-255. doi
Grabchak M., Molchanov S. Limit theorems for random exponentials: the bounded support case // Theory of Probability and Its Applications. 2019. Vol. 63. No. 4. P. 634-647. doi
Manita O. A., Veretennikov A. On convergence of 1D Markov diffusions to heavy-tailed invariant density // Moscow Mathematical Journal. 2019. Vol. 19. No. 1. P. 89-106. doi
Jabir J. M., Profeta C. A stable Langevin model with diffusive-reflective boundary conditions // Stochastic Processes and their Applications. 2019. Vol. 129. No. 11. P. 4269-4293. doi
Molchanov S., Vainberg B. Population dynamics with moderate tails of the underlying random walk // SIAM Journal on Mathematical Analysis. 2019. Vol. 51. No. 3. P. 1824-1835. doi
Cranston M., Molchanov S. On the critical behavior of a homopolymer model // Science China Mathematics. 2019. Vol. 62. No. 8. P. 1463-1476. doi
Kolokoltsov V. The probabilistic point of view on the generalized fractional partial differential equations // Fractional Calculus and Applied Analysis. 2019. Vol. 22. No. 3. P. 543-600. doi
Palamarchuk E. S. On Upper Functions for Anomalous Diffusions Governed by Time-Varying Ornstein-Uhlenbeck Process / Пер. с рус. // Theory of Probability and Its Applications. 2019. Vol. 64. No. 2. P. 209-228. doi
Grabchak M., Molchanov S. The alloy model: Phase transitions and diagrams for a random energy model with mixtures // Markov Processes and Related Fields. 2019. Vol. 25. No. 4. P. 591-613.
Веретенников А. Ю. О слабых решениях сильно вырожденных СДУ // Автоматика и телемеханика. 2020. № 3. С. 1-15.
Veretennikov A. On polynomial recurrence for reliability system with a warm reserve / Cornell University. Series cond-mat "arxiv.org". 2019.
Veretennikov A. On Polynomial Recurrence for Reliability System with a Warm Reserve // Markov Processes and Related Fields. 2019. Vol. 25. P. 745-761.
Veretennikov A., Veretennikova M. On convergence rate for homogeneous Markov chains / Cornell University. Series "Working papers by Cornell University". 2019.
Kelbert M., Chernov A., Shemendyuk A. Fair insurance premium level in connected SIR model under epidemic outbreak / Cornell University. Series "Working papers by Cornell University". 2019. No. 1910.04809v1.
Kelbert M., Moreno-Franco H. A. HJB equations with gradient constraint associated with controlled jump-diffusion processes // SIAM Journal on Control and Optimization. 2019. Vol. 57. No. 3. P. 2185-2213. doi
Frikha N., Konakov V., Menozzi S. Well-posedness of some non-linear stable driven SDEs / Cornell University. Series arxive "math". 2019. No. 1910.05945.
Falaleev A., Konakov V. Convergence of some classes of random flights in Wasserstein distance / Cornell University. Series arxive "math". 2019. No. 1910.03862.
Chernov A., Kelbert M., Shemendyuk A. Optimal vaccine allocation during the mumps outbreak in two SIR centres // Mathematical Medicine and Biology. 2019 doi
Molchanov S., Panov V. Limit theorems for the alloy-type random energy model // Stochastics-An International Journal of Probability and Stochastic Processes. 2019. Vol. 91. No. 5. P. 754-772. doi
Panov V., Samarin E. Multivariate asset-pricing model based on subordinated stable processes // Applied Stochastic Models in Business and Industry. 2019. Vol. 35. P. 1060-1076. doi