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.