Stochastic calculus for fractional Brownian motion and related processes, financial mathematics, theory and statistics of random processes and fields, stochastic differential equations

Previous and Current Research

• Stochastic calculus for fractional Brownian motion

• Financial mathematics

• Theory of multiparameter random processes

• Functional limit theorems for random processes and fields

• Martingales and related processes

• Stochastic integration

• Stochastic differential equations

• Optimal stopping problems

• Statistics of the processes with long-range dependence

Yuliya Mishura is the supervisor of the scientific project titled “Exact formulas, estimates, asymptotic properties and statistical analysis of complex evolutionary systems with many degrees of freedom”. The research is concentrated on the development of the theory of processes with long-range dependence, in particular, mixed fractional and multifractional processes, as well as statistical estimation of parameters in fractional and mixed models and option pricing for assets prices of which follow geometrical Brownian motion with long-range dependence.

The team consists of 15 researchers and 8 PhD students:

7 DSc: V.Knopova, M.Maiboroda, M.Moklyachuk, K.Ralchenko,

                 L.Sakhno, G.Shevchenko, R.Yamnenko

8 PhD: I.Bodnarchuk, O.Borysenko, O.Chernova, V.Golomozi,

                  O.Ragulina, S.Shklyar, V.Zubchenko, T.Yanevych

8 PhD students: D.Avetisian, O.Hopkalo, O. Kharytonova, S.Logvinenko,

                  V.Miroshnichenko, G.Navara, M.Yakovliev, G.Zhelezniak

    We maintain close collaboration with universities of Vilnius, Oslo, Liverpool, Napoli, Dresden and others.

Projects:

2000-2002 - INTAS Grant 99-000-16 “Stochastic Processes and stochastic analysis with statistical applications”, coordinator of the Ukrainian group,

2004-2008 - TEMPUS-TACIS IB-JEP-25054-2004 “Educational Courses for Actuaries and Financial Analysists”, leader of the team of Kyiv National University,

2003-2006 - NATO Grant PST.CLG.980408 “Fractional calculus and related stochastic processes and equations”,

2009-2012 - Research European project Multifractionality, Grant Agreement N.230804, “Multi-parameter Multi-fractional Brownian Motion”, International Research Staff Exchange (IRSES), leader of the team of Kyiv National University.

2018-2020 - Project STORM: “Stochastics for Time-Space Risk Models” with University of Oslo.

2020-2022 - Grant of State Fund for Fundamental Research of Ukraine, project “Estimation of parameters, testing of hypotheses and forecasting in actual stochastic models”, 2020.02/0026

We have made a considerable contribution to stochastic calculus of fractional and multifractional processes, stochastic differential equations and corresponding statistical and financial problems. Particularly, we have investigated classes of multifractional processes, mixed models with long-range dependence and their financial applications. We were the first who obtained general functional limit theorems for multiplicative schemes that are used in financial mathematics.

Now we work on estimation of parameters in mixed models with long and short-range dependence and multifractional models. We consider approximations for fractional Poisson processes. We estimate parameters of fractional and multifractional process and fields with non-Gaussian distributions. We prove functional limit theorems in multiplicative financial schemes with long-range dependence, consider optimal stopping rules, apply Malliavin calculus to financial models.

Future Projects and Goals

·       

• Integral representations of fractional non-Gaussian processes

• Parameter estimation in mixed fractional and multifractional models

• Functional limit theorems in multiplicative financial schemes

• Hedging and optimization of markets with long-range dependence

• Properties of random dynamic systems driven by mixed stochastic differential equations

•  Properties of statistical estimates for multifractional non-Gaussian models

• Approximate methods for mixed stochastic differential equations

• Asymptotic behavior of the solutions of stochastic differential equations

• Optimal stopping problems

Methodological and Technical Expertise

• Stochastic analysis

• Theory and statistics of stochastic processes

• Fractional processes

• Financial mathematics

Selected Publications

• G. Kulinich, S. Kushnirenko, Yu. Mishura. "Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations". Vol.9, Bocconi & Springer Series, Mathematics, Statistics, Finance and Economics, 248 p. - 2020.
• O. Banna, Yu. Mishura, K. Ralchenko, S. Shklyar. "Fractional Brownian Motion. Approximations and Projections". Wiley-ISTE, 288 p. - 2019.
• Yu. Mishura, M. Zili. "Stochastic Analysis of Mixed Fractional Gaussian Processes". ISTE Press - Elsevier, 210 p. - 2018.
• K. Kubilius, Yu. Mishura, K. Ralchenko. "Parameter Estimation in Fractional Diffusion Models". Bocconi & Springer Series, 380 p. - 2017.
• Yu. Mishura. "Stochastic calculus for fractional Brownian motion and related processes". Vol. 1929 of Lecture Notes in Mathematics, Springer Science & Business Media, 410 p. - 2008.
• J. Eisenberg, Yu. Mishura. "Optimising Dividends and Consumption Under an Exponential CIR as a Discount Factor". Mathematical Methods of Operations Research, Vol. 92, pp. 285-309. - 2020.
• Yu. Mishura, K. Ralchenko, S. Shklyar "General Conditions of Weak Convergence of Discrete-Time Multiplicative Scheme to Asset Price with Memory ". Risks, Vol. 8, Iss.1, pp. 1-29. - 2020.
• G. Ascione, Yu. Mishura, E. Pirozzi. "Time-Changed Fractional Ornstein-Uhlenbeck Process". Fractional Calculus and Applied Analysis, Vol. 23, Iss.2, pp. 450- 483. - 2020.
• Yu. Kondratiev, Yu. Mishura, G. Shevchenko. " Limit theorems for additive functionals of continuous time random walks". Proceedings of the Royal Society of Edinburgh Section A: Mathematics, pp. 1-22. - 2020.
• P. Guasoni, Yu. Mishura, M. Rasonyi. "High-Frequency Trading with Fractional Brownian Motion". Finance and Stochastics. - 2020.
• V. Bezborodov, L. Di Persio, Yu. Mishura "Option Pricing with Fractional Stochastic Volatility and Discontinuous Payoff Function of Polynomial Growth". Methodology and Computing in Applied Probability. Vol.21, Iss.1, pp. 331-366. - 2019.
• Yu. Mishura, A. Schied. "On (signed) Takagi–Landsberg functions: pth variation, maximum, and modulus of continuity". Journal of Mathematical Analysis and Applications, Vol. 473, Iss.1, 258-272. - 2019.

Contacts