Stochastic analysis, Stochastic differential equations, Models with long memory, Financial mathematics, Statistics of random processes and fields
Research Fields:
Mathematics
Previous and Current Research
 Stochastic analysis
 Stochastic differential equations
 Fractional Brownian motion
 Fractional and multifractional processes
 Stochastic processes and fields with heavytailed distributions
 Numerical methods for stochastic differential equations
 Financial mathematics
 Optimal stopping
 Statistics of random processes and fields
The research group led by Georgiy Shevchenko focuses on fractional and multifractional processes and fields, statistics of stochastic processes and fields, and financial mathematics.
Concerning the first topic, the selfsimilarity phenomena, or fractal phenomena, are being actively investigated by various research groups throughout the world. The reason of such profound interest to these phenomena is their ubiquity in different areas. The fractal behavior can appear statically, in which case it is usually referred to as fractality, or dynamically, in which case it is called fractionality. Modern concepts of multifractality and multifractionality are further extensions of these notions. They are used to describe the phenomena which are only locally selfsimilar. Again, the locality here may refer to time (fractionality over small time intervals) or to space (fractality in small space domains).
A mixed stochastic differential equation (SDE) uses a combination of Wiener process and a long memory process as its random driver. It makes an example of a process, with is multifractional with respect to scale. The research group has established existence and uniqueness results for such SDEs and studied the properties of solutions: integrability, existence of density with respect to the Lebesgue measure. Numerical methods for mixed SDEs had been established so far mostly in a onedimensional smooth case. The top priority is to extend it to several dimensions and to consider more precise schemes such as the modified Euler scheme.
The research group obtained some preliminary results on statistical estimation in mixed models. The estimators we constructed are far from being optimal, so the first priority is to get more efficient estimators. While mixed stochastic differential equations are now an active research area, statistical methods for them remain almost unexplored. The first priority task is to get estimates for the Hurst parameter of the fractional Brownian motion (or other fractional process) driving the mixed SDE.
The group has constructed some estimators for generalized fractional chisquare processes. It is currently working both on constructing more efficient estimators and on extension of these results to a multifractional case.
The group had studied the problem of nonparametric estimation of a kernel function in a movingaverage representation of infinitedivisible random field. The field is assumed to be observed at some uniform grid, which is growing, and at the same time the distance between points is vanishing.
Future Projects and Goals
 Stochastic partial differential equations with heavytailed noise
 Statistical estimation in multifractional models
 Local properties of chisquare processes
 Statistics for chisquare processes
Selected Publications
Yu. Mishura, G. Shevchenko
Small ball properties and representation results
Stochastic Processes and their Applications, Vol.127, Iss.1 pp. 20  36,  2017
A. Iksanov, Z. Kabluchko, A. Marynych, G. Shevchenko
Fractionally integrated inverse stable subordinators
Stochastic Processes and their Applications, Vol.127, Iss.1 pp. 80  106,  2017
T. Shalaiko, G. Shevchenko
Existence of density for solutions of mixed stochastic equations
Stochastic and Infinite Dimensional Analysis (Editors: Ch. Bernido, M. CarpioBernido, M. Grothaus, T. Kuna, Maria João Oliveira, José Luís da Silva), pp. 281  300,  2016
Mishura Yu., Shalaiko T., Shevchenko G.
Convergence of solutions of mixed stochastic delay differential equations with applications
Appl. Math. Comput. , Vol.257, Iss. pp. 487  297,  2015
Melnikov A., Mishura Yu., Shevchenko G.
Stochastic viability and comparison theorems for mixed stochastic differential equations
Methodol. Comput. Appl. Probab., Vol.17, Iss.1 pp. 169  188,  2015
M. Dozzi, Yu. Mishura, G. Shevchenko
Asymptotic behavior of mixed power variations and statistical estimation in mixed models
Statistical Inference for Stochastic Processes, Vol.18, Iss.2 pp. 151  175,  2015
G.Shevchenko
Mixed fractional stochastic differential equations with jumps
Stochastics: An International Journal of Probability and Stochastic Processes, V. 86,, pp. 203  217,  2014
Shevchenko G., Viitasaari L.
Integral Representation with Adapted Continuous Integrand with Respect to Fractional Brownian Motion
Stoch. Anal. Appl., Vol.32, Iss.6 pp. 934  943,  2014
G.Shevchenko,Shalaiko T.O.
Malliavin regularity of solutions to mixed stochastic differential equations
Statistics and Probability Letters, V. 83, no. 12, pp. 2638  2646,  2013
G.Shevchenko,Mishura Yu.S
Mixed stochastic differential equations with longrange dependence: existence, uniqueness and convergence of solutions
Ñomputers and Mathematics with Applications, V. 64, no. 10, pp. 3217  3227,  2012
G.Shevchenko,Mishura Yu.S. Valkeila E.
Random variables as pathwise integrals with respect to fractional Brownian motion
Stochastic Processes and Their Applications, V. 123, no. 6, pp. 2353  2369,  2012
Ì. Dozzi, G. Shevchenko
Real harmonizable multifractional stable process and its local properties
Stoch. Proc. Appl, Volume 121, Issue 7, pp. 1509  1523,  2011
Y. Mishura, G. Shevchenko
Existence and uniqueness of solution of stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index H º (1/2,1).
Communications in Statistics: Theory and Methods. V.40, ¹ 1920, pp. 3492  3508,  2011
Contacts
Homepage: http://probability.univ.kiev.ua/index.php?page=userinfo&person=zhoraster&lan=en
zhora@univ.kiev.ua
