Georgiy Shevchenko




Georgiy Shevchenko

POSITION
Head of the Department of Probability, Statistics and Actuarial Mathematics

WORK EXPERIENCE
2003–2008
Assistant Professor
Taras Shevchenko National University of Kyiv, Kyiv (Ukraine)

2008–2016
Associate Professor
Taras Shevchenko National University of Kyiv, Kyiv (Ukraine)

2016–Present
Professor
Taras Shevchenko National University of Kyiv, Kyiv (Ukraine)

EDUCATION AND TRAINING

2001
MSc
Taras Shevchenko National University of Kyiv, Kyiv (Ukraine)

2005
PhD
Taras Shevchenko National University of Kiev, Kyiv (Ukraine)

2014
DSc (Habilitation)
Taras Shevchenko National University of Kiev, Kyiv (Ukraine)

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 heavy-tailed 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 self-similarity 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 self-similar. 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 one-dimensional 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 chi-square 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 non-parametric estimation of a kernel function in a moving-average representation of infinite-divisible 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 heavy-tailed noise
  • Statistical estimation in multifractional models
  • Local properties of chi-square processes
  • Statistics for chi-square 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. Carpio-Bernido, 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 long-range 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

̀. 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 19-20, pp. 3492 - 3508, - 2011

Contacts

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zhora@univ.kiev.ua