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# NEGA-$\tilde Q$-REPRESENTATION OF REAL NUMBERS

Authors:
TARAS SHEVCHENKO NATIONAL UNIVERSITY OF KYIV
INTERNATIONAL CONFERENCE
PROBABILITY, RELIABILITY AND
STOCHASTIC OPTIMIZATION
Dedicated to the 90th anniversary of V. S. Koroliuk,
80th anniversary of I. M. Kovalenko,
75th anniversary of P. S. Knopov and
75th anniversary of Yu. V. Kozachenko
April 7–10, 2015, Kyiv, Ukraine
CONFERENCE MATERIALS
ORGANIZED BY
Taras Shevchenko National University of Kyiv
Institute of Mathematics of the National Academy of Sciences of Ukraine
V. M. Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine
National Technical University of Ukraine “Kiev Polytechnical Institute”
Dragomanov National Pedagogical University
Ukrainian Charitable Foundation for Furthering Development of Mathematical Science
PROGRAM COMMITTEE
V. Koroliuk (Honorary Co-Chair), I. Kovalenko (Honorary Co-Chair),
P. Knopov (Co-Chair), Yu. Kozachenko (Co-Chair),
O. Ivanov, O. Klesov, Yu. Kondratiev, K. Kubilius, O. Kukush, N. Leonenko,
R. Maiboroda, E. Orsingher, M. Pratsiovytyj, M. Savchuk, G. Torbin
ORGANIZING COMMITTEE
O. Zakusylo (Co-Chair), A. Samoilenko (Co-Chair),
I. Sergienko (Co-Chair), Yu. Mishura (Vice-Chair),
O. Borisenko, V. Golomozi, V. Knopova, A. Kulik, R. Nikiforov,
V. Radchenko, K. Ral’chenko, O. Ragulina, L. Sakhno, I. Samoilenko,
G. Shevchenko, O. Vasylyk, R. Yamnenko, T. Yanevych, V. Zubchenko
VOLODYMYR SEMENOVYCH KOROLIUK
(TO THE 90TH ANNIVERSARY)
On August 19 there will be 90th anniversary of the outstanding Ukrainian scientist in the ﬁeld of probability theory,
mathematical statistics and cybernetics – Academician of NAS of Ukraine Volodymyr Semenovych Koroliuk.
V. S. Koroliuk was born in Kyiv, where he received a secondary education. Being in the military service, he
graduated from the ﬁrst two courses of Kharkiv University by correspondence, and since 1947 continued his studies at
Taras Shechenko Kyiv University, which he graduated in 1950. Research interests of the future scientist were shaped
by the inﬂuence of Academician B. V. Gnedenko.
Since 1954 he is constantly working at the Institute of Mathematics of NAS of Ukraine (former Academy of Sciences
of USSR), ﬁrst as a research assistant, then since 1956 – as a senior researcher, and since 1960 as the Head of the
Department of probability theory and mathematical statistics. In 1963 he defended his Doctoral thesis “Asymptotical
analysis in boundary problems of random walks”. Since 1966 till 1988 V. S. Koroliuk was the Deputy Director of
the Institute of Mathematics of AS of USSR. In 1967 he was elected a Corresponding member and in 1976 – the
Academician of the AS of USSR.
V. S. Koroliuk is one of the ﬁrst scientists in Ukraine, who assessed the theoretical and practical importance of
semi-Markov processes and attracted the attention of his students for their research and application. The results of
these studies launched a new direction – the theory of asymptotic phase merging and averaging of random processes.
They were summarized in the monographs of V. S. Koroliuk and A. F. Turbin: “Semi-Markov processes and their
applications” (in Russian – 1976), “Mathematical foundations of the state lumping of large systems” (in Russian –
1978, in English – 1993) and a manual “State lumping of large systems” (in Russian – 1978).
In 1980-th V. S. Koroliuk launched another new direction – asymptotic analysis of random evolutions. The research
results of this direction are summarized in the monographs “Stochastic models of systems” (in Russian – 1989, in
Ukrainian – 1993, in English – 1999 – with V. V. Korolyuk as a co-author) and “Semi-Markov random evolutions”
with A. V. Swishchuk as a co-author (in Russian – 1992, in English – 1995).
Since 1990-th years V. S. Koroliuk continued expansion of new asymptotic methods for evolutionary systems with
random perturbations. Long years of creative collaboration in the studying of phase merging of V. S. Koroliuk and
N. Limnios – Professor of Technological University of Compiegne (France) – promoted in 2005 (the 80-th anniversary
of the hero of the day) another monograph: V. S. Koroliuk, N. Limnios “Stochastic systems in merging phase space”
of World Scientiﬁc Publishers.
The mathematical heritage of V. S. Koroliuk covers 22 monographs and about 20 textbooks, most of which are
reprinted in foreign languages; about 280 scientiﬁc articles, about 50 popular science articles and editorial publications
to “Encyclopedia of Cybernetics” (in Russian and Ukrainian), monographs, reference books and scientiﬁc collections.
The outstanding scientist combines fruitful scientiﬁc work, teaching and scientiﬁc-organizational activities. Since
1954 he lectured on the theory of programming, probability theory and mathematical statistics at the Taras Shevchenko
Kyiv University (Faculty of mechanics and mathematics). The “Manual on probability theory and mathematical
statistics” (in Russian) was published in 1978 by his editorship and was repeatedly reprinted in diﬀerent languages. As
a part of a group of famous experts, he was awarded by the USSR State Prize (1978) for the creation of “Encyclopedia
of Cybernetics”. In addition, V. S. Koroliuk was awarded by the Glushkov Prize (1988) and Bogolyubov Prize
(1995). In 1998 he was awarded by the honorary title “Honored Figure of Science and Technics of Ukraine.” In 2002
V. S. Koroliuk was awarded by the Prize of the National Academy of Sciences of Ukraine and Medal in the name of
M. V. Ostorgradsky, and in 2003 – the State Prize of Ukraine in Science and Technology.
Under his supervision, 42 students defended their PhD dissertations, 14 – Doctoral dissertations.
At his age of 90 the Scientist continues active scientiﬁc, educational and organizational work, lives in creative
pursuits and plans. He combines creative activities, lectures and presentations at various international conferences
and in scientiﬁc centers of Italy, Spain, Holland, Germany, France, Switzerland and Sweden. He is actively involved
in organizing and conducting of international conferences. He is the Editor-in-Chief of “Theory of Probability and
Mathematical Statistics”, associate editor of “Ukrainian Mathematical Journal”, “Cybernetics and Systems Analysis”,
“Theory of Stochastic Processes” and other journals.
3
Till some last years before the anniversary V. S. Koroliuk realized his dream: he applied the methods of solving
the singular perturbation problem to the problems of large deviations for random evolutions.
Recent selected publications:
(1) Korolyuk, V. S., Problem of large deviations for Markov random evolutions with independent increments in
the scheme of asymptotically small diﬀusion. Ukrainian Math. J. 62 (5), 2010, 739–747.
(2) Koroliuk, V. S., Markov random evolutions with independent increments in the scheme of asymptotically small
diﬀusion (in Russian). Dopovidi NANU, 2010.
(3) Koroliuk, V., Principle of large deviations for random evolutions (in Russian). Visnyk KNU imeni Tarasa
Shevchenka 25, 2011, 4–6.
(4) Koroliuk, V. S., Large deviation problems for Markov random evolution with independent increments in the
scheme of asymptotically small diﬀusion. Comm. Statist. Theory Methods 40 (19-20), 2011, 3385–3395.
(5) Koroliuk, V. S., Random evolutions with locally independent increments on increasing time intervals (in Rus-
sian). Ukr. Mat. Visn. 8(2), 2011, 220–240, 316; translation in J. Math. Sci. (N. Y.) 179 (2), 2011,
273–289.
(6) Koroliuk, V. S., Dynamic random evolutions on increasing time intervals (in Ukrainian). Teor. Imovir. Mat.
Stat. 85, 2011, 75–83; translation in Theory Probab. Math. Statist. 85, 2012, 83–91.
(7) Koroliuk, V. S., Manca, R., D’Amico G., Storage impulsive processes in the merging phase space. Ukr. Mat.
Visn. 10 (3), 2013, 333–342; translated in Journal of Mathematical Sciences, 2013.
(8) Koroliuk, V. S., Manca, R., D’Amico, G., Storage impulsive processes on increasing time intervals. Teor.
Imovir. Mat. Stat. 89, 2013, 64–74.
(9) Koroliuk, V. S., Samoilenko, I. V., Large deviations for random evolutions in the scheme of asymptotically
small diﬀusion. Modern Stochastics and Applications, Springer Optimization and Its Applications 90, 2014,
201–217.
(10) Koroliuk, V. S., Samoilenko, I. V., Large deviations for storage impulsive processes in the merging phase scheme
(in Ukrainian). Dopovidi NANU 7, 2014, 28–35.
(11) Koroliuk, V. S., Manca, R., D’Amico, G., Stochastic impulsive processes on superposition of two renewal
processes. Ukr. Mat. Visn. 11 (3), 2014, 391–404.
4
IGOR MYKOLAYOVYCH KOVALENKO
(TO THE 80TH ANNIVERSARY)
March 16, 2015 we celebrated 80 years since the birth of the outstanding scientist – mathematician and cyber-
netician – Professor, Doctor of Sciences in Physics and Mathematics, Doctor of Technical Sciences, Academician of
the National Academy of Sciences of Ukraine, Laureate of the State Prizes of the USSR, UkrSSR and Ukraine Igor
Mykolayovych Kovalenko.
I. M. Kovalenko was born in Kyiv on 16.03.1935. His parents Mykola Oleksandrovych Kovalenko (1904–1977)
and Valeriya Volodymyrivna Yavon (1913–1997) were engineers-designers and builders. Childhood of Igor Kovalenko,
including the years of World War II, was held in the village of Sloboda in Chernihiv region, where his grandparents,
Volodymyr Mykhailovych Yavon and Zoya Ivanivna Yavon (Reichardt), worked as teachers.
From 1946 to 1961 Igor Kovalenko lived in Kyiv with his father. Igor Kovalenko married in 1957. With his wife
Olena Markivna Kovalenko (Braga), they had been married for 55 years until her death in 2012. They have two
daughters – Galyna and Yeva.
Igor Kovalenko received his primary education in a school in the village of Sloboda. Then his father, demobilized
from the army, took Igor to one of the best Ukrainian secondary schools named after Ivan Franko (former Pavlo
Galagan College) in Kyiv. After graduating with excellent marks (1952), Igor Kovalenko entered the Mechanics and
Mathematics Faculty of Taras Shevchenko Kyiv State University. On the advice of Y. I. Gikhman (at that time
assistant professor) Igor Kovalenko joined a group of mathematicians. Later, he was among the students, who were
taken care of by the Theory of Probability and Algebra Department, which was headed by academician of AS of the
In student years Igor Kovalenko began his scientiﬁc work: a graduate thesis was performed under supervision of
V. S. Mykhalevych, who was a young teacher at that time. The formation of Igor Kovalenko as a mathematician was
also promoted by professors of the Department L. A. Kaluzhnin, Y. I. Gikhman, O. S. Parasyuk, Y. M. Berezanskii,
The period of postgraduate studies under the supervision of B. V. Gnedenko at the Institute of Mathematics
of the AS UkrSSR was very fruitful scientiﬁcally for Igor Kovalenko. Boris Volodymyrovych directed the eﬀorts
of his postgraduate students (apart from Igor Kovalenko, B. I. Grigelionis, M. V. Yarovytskii, T. P. Marianovych,
S. M. Brodie, A. A. Shakhbazov, T. I. Nasirova et al.) to the problems of queueing and reliability theories. In this
research ﬁeld Igor Kovalenko studied a range of queueing systems with impatient customers. The results of Igor
Kovalenko formed the basis of his PhD thesis (1960). During the subsequent year and a half of work in the Institute of
Mathematics (1960–1961) Igor Kovalenko discovered the so-called invariance (or insensitivity) criterion of the queueing.
This result played its role in the early development of the insensitivity theory in the USSR and beyond.
Together with B. V. Gnedenko he wrote the book [1].
Along with his main teacher B. V. Gnedenko, Igor Kovalenko considers as his closed teachers the academicians
V. S. Mykhalevych (1930–1994) and V. S. Koroliuk (b. 1925). Yes, V. S. Koroliuk engaged him in the small parameter
method, in which he himself received great results.
The advices of academician A. N. Kolmogorov, especially his fruitful critical remarks, and of academician Yu. V. Lin-
nik played a signiﬁcant role in the formation of Igor Kovalenko as a scientist.
From his teacher Gnedenko Igor Kovalenko adopted his tendency to ﬁnding applied problems in order to solve
them with the help of mathematical theory. So it was very natural for Igor that in January 1962 he moved to Moscow
to work at the Institute “SNII-45” of the USSR Defense Ministry to study the challenges facing the systems reliability.
There he worked for almost ten years – until mid-1971. The probabilistic model of the very complex defense system
was created and investigated; as a by-product of this engineering study emerged the asymptotic method of reliability
analysis of complex systems [2].
In “SNII-45” the responsible for science was the famous scientist, N. P. Buslenko. He got Igor Kovalenko involved
to the development of general models of complex systems. In this ﬁeld, Igor Kovalenko proposed the concept of the
so-called piecewise linear aggregates (PLA) and piecewise linear Markov processes (PLMP). Elements of the PLA
theory were included in the monograph [3], while PLMP formed the basis of multidimensional queueing processes in
the monograph [4]. In 1964 Igor Kovalenko defended his doctoral thesis in technical sciences.
5
We should recall the Moscow student of Igor Kovalenko, who was distinguished by fundamental results in the
theory of network services, – V. A. Ivnitskii. His latest books are [5,6].
The probabilistic combinatorics was another research ﬁeld of Igor Kovalenko in the Moscow period. His second
doctoral thesis (1970) was mainly devoted to the theory of random Boolean equations that had practical applications.
For this thesis Igor Kovalenko received his degree of Doctor of Sciences in Physics and Mathematics.
B. V. Gnedenko, who moved to Moscow in 1960, organized a seminar on queueing and realibility theories in Mechan-
ics and Mathematics Faculty of Moscow State University. The seminar was the real scientiﬁc school. B. V. Gnedenko
recruited A. D. Solovyov, Yu. Рљ. Belyayev, and Igor Kovalenko as co-leaders of the seminar.
By the invitation of Academician V. M. Glushkov Igor Kovalenko returned back to his native Kyiv in mid-1971.
He headed the Department of Mathematical Methods of Reliability Theory of Complex Systems at the Institute of
Cybernetics of the AS of UkrSSR. He still works at this position, that is for almost of 44 years. (The current name of
the institute: V. M. Glushkov Institute of Cybernetics of NAS of Ukraine).
The traditional subject of the Department research – methods for reducing the variance of estimates of systems
reliability – is leaded by a disciple of Igor Kovalenko, a Corresponding Member of the NAS of Ukraine M. Yu. Kuznetsov.
Joint monographs by Igor Kovalenko and M. Yu. Kuznetsov: [7,8].
Another worthy contribution relating mathematical reliability theory was done by an employee of the Department
L. S. Stoikova [9]. She developed a method for evaluating upper and lower limits of system reliability in the case of
incomplete information about the characteristics of the system.
In the Institute of Cybernetics Igor Kovalenko was also engaged in probabilistic combinatorics.
In mid-1960 A. M. Kolmogorov formulated the limit distribution problem of the random Boolean determinant.
Igor Kovalenko solved this problem, see [10]. Signiﬁcant generalization of this result was developed primarily by
A. O. Levytska, see [11–12]. A new approach to this problem was introduced by O. M. Oleksiychuk [13].
New tasks for himself Igor Kovalenko found during numerous international visits, starting from 1994 (Switzerland,
United Kingdom, Germany, USA, Denmark, Israel, etc.). So, he found the ﬁrst rigorous proof of the conjecture of
D. Kendall from stochastic geometry [14], and found the estimation of the random cell area (Crofton cell) for Poisson
point process [15].
Finally, let us note the works of Igor Kovalenko (mainly written together with O. V. Koba) [16–18] on the retrial
queues. They obtained stability conditions for the systems, which generalized the systems investigated by L. Lakatos
(Iotvos Lorand University, Budapest, Hungary).
Let us note also the following professorial activity of Igor Kovalenko. In Taras Shevchenko National University of
Kyiv Igor Kovalenko worked as a part-time professor during 17 academic years, beginning with 1971/1972. Several
years he worked at Moscow Institute of Electronic Engineering, KVIRTU PVO and NTUU “KPI” (in the latter Igor
Kovalenko was a Dean of the Faculty of Applied Mathematics).
He published several co-authored textbooks, among which [19–21].
Igor Kovalenko was the supervisor of 30 PhD theses. He considers 10 Doctors of Sciences to be his worthy disciples.
These are the scientists: A. A. Alekseiev, V. A. Ivnitsky, O. V. Koba, M. Yu. Kuznetsov, A. O. Levytska, V. I. Masol,
O. M. Nakonechy, M. M. Savchuk, L. S. Stoikova, V. D. Shpak.
References
[1] Gnedenko, B. V., Kovalenko, I. N. Lectures on Queueing Theory 1-3 (in Russian). KVIRTU, Kyiv, 1963, 316 p.
[2] Kovalenko, I. N., Rare events in queueing systems. Queueing Systems 16 (1), 1994, 1–49.
[3] Buslenko, N. P., Kalashnikov, V. V., Kovalenko, I. N. Lectures on Complex Systems Theory (in Russian). Sov. Radio, Moscow, 1973,
439 p.
[4] Gnedenko, B. V., Kovalenko, I. N. Introduction to Queueing Theory (in Russian), 6th Ed. URSS, Moscow, 2012, 397 p. Translation:
Gnedenko, B. V., Kovalenko, I. N. Introduction to Queueing Theory. Birkh¨auser, Boston, 1989, 315 p.
[5] Ivnitsky, V. A. Theory of Random Input Flow with Applications to the Transcient Queueing Systems and Networks (in Russian).
Palmarium, Germany, 2012.
[6] Ivnitsky, V. A. Recurrent Modeling of Discrete-Continuous Markov Processes (in Russian). Palmarium, Germany, 2013.
[7] Kovalenko, I. N., Kuznetsov, N. Yu. Methods for Calculating the Highly Reliable Systems (in Russian). Radio and Communication,
Moscow, 1978, 176 p.
[8] Kovalenko, I. N., Kuznetsov, N. Yu., Pegg, Ph. The Theory of Reliability of Time-Dependent Systems with Practical Applications.
Wiley, Chichester, 1997.
[9] Stoikova, L. S., Generalized Chebyshev inequality and its application in the mathematical theory of reliability (in Russian). Cybernetics
and Systems Analysis 3, 2010, 139–143.
[10] Kovalenko, I. N., Limit distribution of the number of solutions of random systems of linear equations in the class of Boolean functions
(in Russian). Probability Theory and its Applications 12 (1), 1967, 51–61.
[11] Kovalenko, I. N., Levitskaya, A. O., Savchuk, M. M. Selected Problems of Probabilistic Combinatorics (in Russian). Naukova Dumka,
Kyiv, 1986, 223 p.
[12] Levitskaya, A. O., Solving the problem of invariance of probability characteristics of random consistent system of nonlinear equations
over a ﬁnite commutative ring with unit (in Russian). Cybernetics and Systems Analysis 3, 2010, 28–41.
[13] Aleksiichuk, A. N., On the uniqueness of the moment problem in the class of q-distributions (in Russian). Discrete Mathematics
10 (1), 1998, 96–110.
[14] Kovalenko, I. N., A proof of the conjecture of David Kendall on the form of random polygons (in Russian). Cybernetics and Systems
Analysis 4, 1997, 3–11.
[15] Kovalenko, I. N., On certain random polygons of large areas. Journal of Applied Mathematics and Stochastic Analysis 11 (3), 1998,
369–376.
[16] Koba, E. V., On a retrial queueing system with a FIFO queueing discipline. Theory of Stochastic Processes 24 (8), 2002, 201–207.
[17] Koba, O. V., Kovalenko, I. N., On the classiﬁcation of queueing systems with retrials (in Russian). Cybernetics and Systems Analysis
3, 2010, 84–91.
[18] Kovalenko, I. N., About two-cycle queueing system (in Russian). Cybernetics and Systems Analysis 1, 2015, 59–64.
[19] Kovalenko, I. N., Filippova, A. A. Probability Theory and Mathematical Statistics (in Russian), 2nd ed. Higher School, Moscow, 1982,
256 p.
[20] Kovalenko, I. N., Sarmanov, O. V. A Short Course in the Theory of Stochastic Processes (in Russian). Kyiv, Vyshcha Shkola, 1978,
264 p.
[21] Ivchenko, G. I., Kashtanov, V. A., Kovalenko, I. N. Queueing Theory (in Russian), 2nd ed. URSS, Moscow, 2012, 304 p.
6
PAVLO SOLOMONOVYCH KNOPOV
(TO THE 75TH ANNIVERSARY)
Pavlo Solomonovych Knopov is the famous and respected scientist in the ﬁeld of computer science, statistical
decision theory and stochastic systems optimal control. His scientiﬁc research is related to new problems of inaccurate
information processing under conditions of incomplete data for the purpose of recognition, identiﬁcation of object
states and its control, mathematical problems of risk theory and its applications in various ﬁelds of economy and
technology. His works are widely recognized by both domestic and foreign experts.
Doctor of Sciences (1986), Professor (1998), Corresponding member of the National Academy of Sciences of Ukraine
(2012), Head of the Department of Mathematical Methods of Operations Research of V. M. Glushkov Institute of
Cybernetics of the National Academy of Sciences of Ukraine, Laureate of the State Prize of Ukraine in Science and
Technology (2009) and V. M. Glushkov Prize (1997), awarded by Certiﬁcate of Honour of the Presidium of the National
Academy of Sciences of Ukraine and by the Diploma of the Verkhovna Rada of Ukraine.
P. S. Knopov was born on May 21, 1940 in Kyiv. He graduated from high school number 21 in 1957. Then he
entered the Mechanics and Mathematics Faculty of Kyiv State University named after T. G. Shevchenko, and graduated
in 1962 with specialty “Mathematics”. The outstanding Ukrainian scientists A. V. Skorokhod, Yu. M. Berezanskyi,
V. S. Mikhalevich, L. A. Kaluzhnin, V. S. Koroliuk, I. M. Kovalenko, M. Y. Yadrenko were his teachers.
The same year he started working in V. M. Glushkov Institute of Cybernetics of the NAS of Ukraine and passed
successively positions from an engineer to the Head of Department. At that time on the initiative of V. S. Mikhale-
vich the joint laboratory of Institute of Cybernetics, Institute of Mathematics and Kyiv National Taras Shevchenko
University was established under the direction of the reputable scientist in the ﬁeld of statistical decision theory
A. Ya. Dorogovtsev to study actual theoretical and applied problems of statistics of random processes and ﬁelds.
The cooperation with A. Ya. Dorogovtsev, as well as with the prominent scientists in the ﬁeld of optimization theory
V. S. Mikhalevich, Yu. Ermoliev, N. Z. Shor and other scientists of the Institute of Cybernetics greatly inﬂuenced
P. S. Knopov’s subsequent scientiﬁc work.
He has been working as a part-time Professor of Applied Statistics Department of the Faculty of Cybernetics, Taras
Shevchenko National University of Kyiv for over 30 years.
P. S. Knopov is an author of 11 monographs and of over 200 research papers published in reputable international
and national journals. Among the monographs there are “Limit theorems for Stochastic Programming Processes”
(Naukova Dumka, 1980, co-authors Yu. M. Kaniovskyy, Z. V. Nekrylova, in Russian), “Stochastic System Parameters
Optimal Estimates” (Naukova Dumka, 1981, in Russian), “Empirical estimates in stochastic optimization and identiﬁ-
cation” (Kluwer Akademic Publishers, 2002, co-author E. J. Kasitskaya), “Control of spatially structured processes and
random ﬁelds with applications” (Springer, 2006, co-authors R. K. Chornei, H. Daduna), “Simulation and Optimization
Methods in Risk and Reliability Theory” (Nova Science Publishers, 2009, eds. P. S. Knopov and P. M. Pardalos), “Re-
gression Analysis under a Priory Parameter Restriction” (Springer, 2012, co-author A. S. Korkhin), “Estimation and
Control Problems for Stochastic Partial Diﬀerential Equations” (Springer, 2013, co-author O. N. Deriyeva), “Robust
Risk Estimation” (V. M. Glushkov Institute of Cybernetics by NAS of Ukraine, 2008, co-authors O. M. Holodnikov,
V. A. Pepeliaev, in Ukrainian).
He made talks at many international conferences, including the World Congress of Mathematicians in Berlin (1998),
the European Congress of Mathematicians in Stockholm (2004), the 16th International Congress on Mathematical
Programming (Lausanne 1997), the International Conference of Stochastic Programming (Columbia University, USA,
1998), International Conference on Risk, SAR and SRA – Europe Annual Conference (London, 2000) and many others.
His research interests include diverse branches of statistical theory of random processes and ﬁelds, the modern
theory of stochastic optimization, forecasting and optimal control of stochastic dynamic multicomponent systems.
Robust statistical methods of identiﬁcation, statistical methods of unknown parameters estimation in the lack of
a priori information, the theoretical foundations of optimal control of dynamic multicomponent stochastic systems,
numerical methods for ﬁnding optimal estimates for complex cybernetic systems that operate in conditions of risk and
uncertainty, developed by P. S. Knopov are widely used in risk estimation in various problems of nuclear energetics,
economy, ecology, geology, recognition theory, reliability theory, the theory of stocks, and so on.
7
He received grants from the Royal Society (UK), London Mathematical Society (UK), DFG (German Foundation
for Basic Research), University of Bolzano (Italy), etc.
He was the manager and executive in charge of many international projects: NATO projects (together with
the University “Florida”, USA), STCU (together with the University “John Washington University”), INTAS, the
company “Intel”, International Foundation SIU (Norway), scientiﬁc and applied research projects carried out by orders
of the Ministry of Education of Ukraine, the Ministry of Defence of Ukraine, a number of other ministries and
organizations are among them. In these projects important problems related to the risk of accidents at hazardous
enterprises estimation in the presence of small statistical samples are resolved. Fundamentally new methods for
reliability parameter estimation for systems based on the original fundamental results in the theory of estimation
in conditions of insuﬃcient statistical information were proposed. The important problem of identifying bottlenecks
that occur in the production of computer equipment and component leading to the output of defective products was
investigated in the joint project with the company “Intel” (USA). Its diﬃculty was caused by the situation when the
very limited time period is available for making optimal decision based on extremely large volumes of observations.
The proposed decision was based on developed methods of control and estimation using modern methods of remote
sensing.
P. S. Knopov devotes a lot of time to scientiﬁc personnel training. A large number of doctors and candidates of
sciences prepared their theses under his supervision. He has been being a member of the Specialized Scientiﬁc Council
of Thesis Defense at V. M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine for many
years.
8
YURIY VASYLIOVYCH KOZACHENKO
(TO THE 75TH ANNIVERSARY)
Doctor of Sciences in Physics and Mathematics, Laureate of the State Prize of Ukraine in Science and Technology,
Honored Personality of Science and Technology of Ukraine, Honored Professor of Taras Shevchenko National University
of Kyiv, Honored Doctor of Uzhgorod National University, Professor Yuriy Vasyliovych Kozachenko is the Dear Teacher
and guide in the world of science for hundreds of students and dozens of postgraduate students.
Yuriy Kozachenko was born in 1940 in Kyiv. In 1963 he graduated from Taras Shevchenko National University
of Kyiv with specialty “Probability Theory and Mathematical Statistics”. His postgraduate study in the Institute of
Mathematics of Academy of Sciences of the Ukrainian SSR was interrupted in 1964–1965 years by military service in
cosmic forces. In 1968, in the Institute of Mathematics of the Ukrainian SSR under supervision of Mykhailo Yadrenko
he defended his thesis for scientiﬁc degree “Candidate of Sciences in Physics and Mathematics” in speciality 01.01.05 –
“Theory of Probability and Mathematical Statistics” titled “On the uniform convergence of stochastic integrals, series
and properties of continuous random ﬁelds”.
Since 1967 he has been working at Taras Shevchenko National University of Kyiv at the Department of Probability
Theory and Mathematical Statistics (since 2009 – Department of Probability Theory, Statistics and Actuarial Mathe-
matics) of Mechanics and Mathematics Faculty. In 1974–1975 he worked at the Institute of Oil and Gas in Budermes
(Algeria).
In 1985, Yuriy Kozachenko defended his doctoral thesis on “Random processes in Orlicz spaces. Properties of
trajectories, convergence of series and integrals”. During the years 1998–2003 he headed the Department of Probability
Theory and Mathematical Statistics of Taras Shevchenko National University of Kyiv. His main scientiﬁc research
interests relate to the study of the properties of random processes in various functional spaces, simulation and statistics
of random processes, the theory of wavelet expansions of random processes. Yuriy Kozachenko is one of the founders
of the theory of subGaussian and ϕ-subGaussian random processes, random processes from Orlicz spaces. He initiated
a research aimed at determination of accuracy and reliability of computer simulation of random processes.
Yuriy Kozachenko is the author of over 300 scientiﬁc papers, several textbooks and seven monographs includ-
ing “Metric characteristics of random variables and processes” (1998, together with V. V. Buldygin), which was
reprinted in English in 2000 by American Mathematical Society, “Simulation of random processes” (1999, together
with A. O. Pashko), “Simulation of random processes and ﬁelds” (2007, together with I. V. Rozora and A. O. Pashko),
“Boundary problems of mathematical physics with random factors” (2008, together with G. I. Slyvka-Teleschak and
B. V. Dovhay), “ϕ-subGaussian random processes” (2008, together with O. I. Vasylyk and R. E. Yamnenko).
As a scientiﬁc supervisor Yuriy Kozachenko have prepared 38 candidates of sciences. Later, one of them Rostyslav
Maiboroda defended doctoral thesis. In addition, for three more scientists – A. P. Yurachkivskyi, O. O. Kurchenko and
I. K. Matsak – Yuriy Kozachenko was a scientiﬁc consultant when they were writing their doctoral theses. His students
continue active research activities at leading universities and institutions around the world. He generously shares his
deep and wide-ranging knowledge with students, postgraduate students, doctoral students, friends and colleagues.
Yuriy Kozachenko supports numerous international links with known scientists from Australia, the UK, Italy,
Canada, USA, Finland, Croatia, Sweden and other countries. Many times he was invited to participate in research
programs in Australia, Italy, Croatia, Finland, Sweden. Yu. Kozachenko conducted research within the framework
of several international projects funded by grants from NATO, TEMPUS-TACIS programme of European Union
and others. He leaded many contractual themes, particularly with design bureaus of State enterprize of a special
instrumentation “Arsenal” and Antonov Aeronautical Scientiﬁc-Technical Complex.
Yuriy Kozachenko won many prizes for scientiﬁc achievements. In 2003 he was awarded by the State Prize of Ukraine
in Science and Technology; in 2005 he was awarded by the title of “Honorary Doctor of Uzhgorod National University”,
in 2009 – by the title of “Professor Emeritus of Taras Shevchenko National University of Kyiv”, in 2010 – by the title of
“Honored Personality of Science and Technology of Ukraine”, and in 2013 he won M. M. Krylov prize of the Presidium
of National Academy of Sciences of Ukraine. Yuriy Kozachenko conducts active scientiﬁc, organizational, educational
and social activities, he is Deputy Chairman of the Specialized Scientiﬁc Council of the Faculty of Mechanics and
Mathematics, a member of the editorial boards of ﬁve academic journals, in particular, he is Deputy Editor-in-Chief
of the scientiﬁc journal “Theory of Probability and Mathematical Statistics’.
9
Students with pleasure attend normative and special courses that he inspiringly teaches at the Faculty of Me-
chanics and Mathematics of the Taras Shevchenko National University of Kyiv, in particular “Theory of Probability
and Mathematical Statistics”, “Generalized Fourier analysis”, “Simulation of random processes”, “Theory of random
processes from Orlicz spaces”, “Wavelet analysis” and others.
The main scientiﬁc works of Yuriy Kozachenko over the past four years:
(1) Kozachenko, Yu. V., Zatula, D. V., Lipschitz conditions for random processes from the Banach spaces Fψ(Ω)
of random variables (in Ukrainian). Dopovidi Natsionalnoi akademii nauk Ukrainy / Reports of the National
Academy of Science of Ukraine 9, 2014, 19–24.
(2) Kozachenko, Yu., Mlavets, Yu., Stochastic processes from Fψ(Ω) spaces. Columbia, International Publishing
Contemporary Mathematics and Statistics 2(1), 2014, 55–75.
(3) Kozachenko, Yu., Slyvka-Tylyshchak, A., The Cauchy problem for the heat equation with a random right part
from the space Subϕ(Ω). Applied Mathematics 5, 2014, 2318–2333.
(4) Kozachenko, Yu., Sergienko, M., The criterion for hypothesis testing on the covariance function of a Gaussian
stochastic process. Monte Carlo Methods and Applications 20 (2), 2014, 137–144.
(5) Kozachenko, Yu., Pashko, A., Accuracy of simulations of the Gaussian random processes with continuous
spectrum. Computer Modelling and New Technologies 18 (3), 2014, 7–12.
(6) Kozachenko, Yu., Slyvka-Tylyshchak, A., On the increase rate of random ﬁelds from the space Subϕ(Ω) on
unbounded domains. Statistics, Optimization and Information Computing 2(2), 2014, 79–92.
(7) Kozachenko, Yu., Olenko, A., Polosmak, O., Uniform convergence of compactly supported wavelet expansions of
Gaussian random processes. Communication in Statistics, Theory and Methods 43 (10–12), 2014, 2549–2562.
(8) Kozachenko, Yu., Yamnenko, R., Buchmitch, D., Generalized sub-Gaussian fractional Brownian queueing
model. Queueing Systems 77, 2014, 75–96.
(9) Kozachenko, Yu., Slyvka-Tylyshchak, A., The Cauchy problem for the heat equation with random right side.
Random Oper. and Stoch. Equation 22 (1), 2014, 53–64.
(10) Kozachenko, Yu., Yamnenko, R., Application of ϕ-sub-Gaussian Random Processes in Queueing Theory.
Springer Optimization and its Applications 90, Modern Stochastics and Application, 2014, 21–38.
(11) Kozachenko, Yu., Ianevych, T., Some goodness of ﬁt tests for random sequences. Lithuanian Journal of
Statistics 52 (1), 2013, 5–13.
(12) Kozachenko, Yu., Kurchenko, O., Synyavska, O., Levy-Baxter theorems for one class of non-Gaussian random
ﬁelds. Monte Carlo Methods and Applications 12 (3), 2013, 171–182.
(13) Kozachenko, Yu., Olenko, A., Polosmak, O., On convergence of general wavelet decompositions of nonstationary
stochastic processes. Electronic Journal of Probability 18 (69), 2013, 1–21.
(14) Kozachenko, Yu., Giuliano-Antonini, R., Tien-Chung, Hu, Volodin, A., On application of ϕ-subgaussian tech-
nique to Fourie analysis. Journal of Mathematical Analysis and Application 408, 2013, 114–124.
(15) Kozachenko, Yu., Olenko, A., Polosmak, O., Convergence rate of wavelet expansions of Gaussian random
processes. Communication in Statistics – Theory and Methods 42 (21), 2013, 3853–3872.
(16) Kozachenko, Yu. V., Mlavets, Yu. Yu., Banach spaces Fψ(Ω) of random variables (in Ukrainian). Theory of
Probability and Mathematical Statistics 86, 2012, 92–107.
(17) Kozachenko, Yu., Kurchenko, O., Levy-Baxter theorems for one class of non-Gaussian stochastic processes.
Random Operators and Stochastic Equations 19 (4), 2011, 21–34.
(18) Kozachenko, Yu., Rozora, I., Turchyn, Ye., Property of some random series. Communications in Statistics –
Theory and Methods 40 (19–20), 2011, 3672–3683.
(19) Kozachenko, Yu., Pogoriliak, O., Simulation of Cox processes driven by random Gaussian ﬁeld. Methodology
and Computing in Applied Probability 13, 2011, 511–521.
(20) Kozachenko, Yu., Mlavets, Yu., Probability of large deviations of sums of random processes from Orlicz space.
Monte Carlo Methods and Applications 17 (2), 2011, 155–168.
(21) Kozachenko, Yu., Olenko, A., Polosmak, O., Uniform convergence of wavelet expansions of Gaussian random
processes. Stochastic Analysis and Applications 29 (2), 2011, 169–184.
Monographs, textbooks and tutorials:
(1) Kozachenko, Yu. V., Pogoriliak, O. O., Tegza, A. M. Simulation of Gaussian Random Processes and Cox
Processes (in Ukrainian). Karpaty, Uzhgorod, 2012.
(2) Dariychuk, I. V., Kozachenko, Yu. V., Perestiuk, M. M. Random Processes from Orlicz Spaces (in Ukrainian).
Zoloti Lytavry, 2011, 212 p.
(3) Vasylyk, O. I., Kozachenko, Yu. V., Yamnenko, R. E. ϕ–subGaussian Random Processes (in Ukrainian). VPC
“Kyivskyi universitet”, Kyiv, 2008, 231 p.
(4) Dovhay, B. V., Kozachenko, Yu. V., Slyvka-Teleschak, G. I. Boundary Problems of Mathematical Physics
with Random Factors. VPC “Kyivskyi universitet”, Kyiv, 2008, 173 p.
(5) Kozachenko, Yu. V., Pashko, A. O., Rozora, I. V. Simulation of Random Processes and Fields (in Ukrainian).
(6) Kozachenko, Yu. V., Pashko, A. O. Simulation of Random Processes (in Ukrainian). VPC “Kyivskyi univer-
sitet”, Kyiv, 1999, 223 p.
(7) Buldygin, V. V., Kozachenko, Yu. V. Metric Characterization of Random Variables and Random Processes
(Transl. from the Russian). Translations of Mathematical Monographs 188. Providence, RI: AMS, American
Mathematical Society xii, 2000, 257 p.
(8) Dovhay, B. V., Kozachenko, Yu. V., Rozora, I. V. Simulation of Random Processes in Physical Systems (in
(9) Kozachenko, Yu. V. Lecture Notes on Wavelet Analysis (in Ukrainian). TViMS, Kyiv, 2004, 147 p.
10
ACT U ARIAL AN D F I NAN CI AL M AT HEM AT I CS
ESTIMATES OF THE PROBABILITY OF BANKRUPTCY
IN THE CASE OF “HEAVY TAILS“
A. Bilynskyi1, O. Kinash2
Classical risk theory assumes that large insurance claims and, therefore, large insurance payments are rare, with
exponentially small probabilities. This scheme is called “model with small payments“.
However, many situations are related to extreme events. Due to this fact, the true size of payments is more
adequately represented by random variables distributed with “heavy tails“, which include Pareto type distributions.
In this case, the total payments are determined by the maximum individual claim.
Note that the class of distributions with “heavy tails“ is wide enough. In particular, it includes so called subexpo-
nential distributions, which were introduced by Chistyakov [1] in the context of the theory of branching processes.
Subexponential distributions and their role in risk theory are discussed, in particular, in the works of Von Bahr [2],
Thorin and Wikstad [3], and Embrechts and Veraverbeke [4].
Review of the theory of subexponential distributions and applications in risk theory in suﬃcient detail is considered
in work of Zinchenko [5].
The authors examined the estimates of the probability of bankruptcy for a number of subexponential distributions
belonging to the Sclass, which deﬁnition can be found in [5, p. 189].
References
[1] Chistyakov, V. P., A theorem on sums of irv and its applications to branching processes. Theor. Probability Appl. 9, 640–648.
[2] Von Bahr, B., Asymptotic ruin probabilities when exponential moments do not exist. Scand. Actuarial J. 1, 1975, 6–10.
[3] Thorin, O., Wikstad, N., Calculation of ruin probabilities when the claim distribution is lognormal. Astin Bulletin 9, 1976, 231–246.
[4] Embrechts, P., Veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance:
Math. Econ. 1(1), 1982, 55–72.
[5] Zinchenko, N. M. Mathematical Methods in Risk Theory: a Tutorial. Kyiv University, 2008.
Ivan Franko National University of Lviv, Universytetska Str. 1, Lviv 79000, Ukraine
SOME CRITERIA FOR OPTIMAL FINANCIAL PORTFOLIO
I. Didmanidze1, T. Kokobinadze2, Z. Megrelishvili3, G. Kakhiani4
The aim of the report is the development of the recommendations for the investors to decide in which securities to
invest the capital and which quantity. The following problems of optimization of ﬁnancial portfolio are considered.
There are nkinds of securities (government securities, municipal bonds, corporation stocks and etc.) with known
mathematical expectations and with known covariance matrix. Leaving for the investor the choice of average eﬃciency
and helping him in this case to minimize the variance, we get the problem of the minimization of the quadratic form
n
P
i=0
bij xixj, where bij =cov(Ri, Rj)and xirepresent the shares of distributed capital.
With the help of Lagrangian function the problem of ﬁnding a conditional extremum is reduced to the problem of
ﬁnding of an unconditional extremum. If in the process of solution some xibecomes zero on negative, the appropriate
securities are excluded from the portfolio and the problem should be solved again.
In the process of buying shares the investor can do the investment without risk. He should combine the risk and
risk-free parts of a portfolio in order to minimize the variance at chosen level of the average eﬃciency. The structure
of securities of the portfolio should repeat the structure of the large market of these securities. The investor can only
vary the share of risk-free securities in his portfolio.
The variance of the portfolio is equal to (1/n)average variance plus (1 1/n)average covariance, that is why if the
shares are not very strongly correlated with each other, the variance of the portfolio reduces with the growing amount
of securities in the portfolio.
Every investor seeks to get an optimal portfolio of risk securities, but the share of the risk-free part of the investment
he deﬁnes by the maximization of mean value of the quadratic function of the utility K·mRai·K2·σ2
R, where Kis
investor’s capital, Ris the casual eﬃciency at chosen structure of the portfolio, coeﬃcients mR=M(R),σ2
R=D(R),
aideﬁne the investor’s tendency to risk, namely, when aiis small, then the tendency to risk is high.
By using approach described above, the authors created and successfully tested software simulations of the ﬁnancial
portfolio. The simulation was carried out on the base of the data of the time series of issuers of the international
share market. The testing shows the high eﬃciency of the used methods and their stability towards changes of the
fundamental data leading to instability in the market. Given methods are very eﬀective in the connection with the
neural network models of forecasting.
References
[1] Kolemaev, V. A. Mathematical Economics (in Russian). UNITI, Moscow, 2002.
[2] Malikhin, V. I. Financial Mathematics (in Russian). UNITI, Moscow, 2003.
Batumi Shota Rustaveli State University, Ninoshvili Str. 35, Batumi 6010, Georgia
11
EUROPEAN OPTION HEDGING IN THE MODEL DESCRIBED
BY GAUSSIAN MARTINGALES WITH DISORDER
O. A. Glonti1, Z. J. Khechinashvili2
On the ﬁltered probability space (Ω,F,(Fn)0nN, P )consider stochastic process in discrete time as evolution of
risky asset price
Sn=S0exp{I(θn)M(1)
n+I(θ < n)M(2)
n}, n = 1, ..., N , (1)
where S0>0is a constant, (M(1)
n,Fn), M (1)
0= 0 and (M(2)
n,Fn), M (2)
0= 0 are independent Gaussian martingales.
θis a random variable which takes values 1,2, ..., N, with known probabilities pi=P(θ=i), i = 1, N . The vector
(M(1), M (2))is independent of θand I(A)is an indicator of A∈ F.
In this model we have obtained optimal in mean square sense hedging strategy for the European contingent claims
f(SN). So, that in the special class of admissible strategies πn= (γn, βn), n 1, N we minimize
E[f(SN)Xπ
N]2,(2)
where Xπ
Nis the capital value at terminal moment N.
Theorem 1. In the model (1) of price evolution, optimal in sense (2) strategy is
γn=E[Fn(∆SnSn1(ehM(1)in
2PN
i=nPn
i+ehM(2)in
2Pn1
i=1 Qn
i1))/FS
n1]
E[(∆SnSn1(ehM(1)in
2PN
i=nPn
i+ehM(2)in
2Pn1
i=1 Qn
i1))2/FS
n1]
,
βn=
n
X
i=1
γiSi1(1 ehM(1)ii
2
N
X
j=i
Pi
jehM(2)ii
2
i1
X
j=1
Qi
j)
n
X
i=1
Si1γi.
where Fn=E[f(SN)/FS
n],FS
n=σ{Si, i n},Pn
iand Qn
iare given in [1].
References
[1] Glonti, O., Khechinashvili, Z., Geometric Gaussian martingales with disorder. Theory of Stochastic Processes 16 (32), 1, 2010, 44–48.
Ivane Javakhishvili Tbilisi State University, University Str. 13, Tbilisi, Georgia
The ﬁrst author is supported by Shota Rustaveli National Science Foundation Grant No. FR/308/5-104/12.
COUPLING MOMENT FOR TIME-INHOMOGENEOUS MARKOV CHAINS
V. V. Golomoziy1, N. V. Kartashov2
In this presentation, we consider conditions which satisfy ﬁniteness of the expectation of the ﬁrst coupling moment
for two independent discrete time-inhomogeneous Markov chains. In order to state a main result, some deﬁnitions
have to be made. Let us deﬁne renewal intervals (θl
k), l ∈ {0,1}for a Markov chain (Xt):
θl
0= inf{t0 : Xt= 0}, θl
m= inf{t>θm1:Xt= 0}, m > 1,
and probabilities gt,l
n=P{θl
k=n|τl
k1=t}, l = 1,2, n 0.We agree that gt,l
0=P{θl
k= 0|τl
k1=t}= 0, where
τl
m=
m
P
k=0
θl
k. Variables θl
k,k1are understood as a renewal step and θl
0is understood as a delay.
We say that T > 0is a coupling (or simultaneous hitting) moment if T= min{t > 0 : m, n :t=τ1
m=τ2
n}.
Our goal is to ﬁnd conditions which guarantee that T < a.s. and E[T]<. We denote as u(t,l)
na renewal
sequence for the process τl:
u(t,l)
0= 1, u(t,l)
n=
n
X
k=0
u(t,l)
kgt+k,l
nk.
Theorem 1. Assume that in notations introduced before:
1) The set of random variables θl(t)is uniformly integrable (in other words, the family of distributions gt,l
nis
uniformly integrable).
2) There exists a constat γ > 0and positive integer n0>0such that for all t, l and nn0:u(t,l)
nγ.
Then the coupling moment is integrable: E[T]<.
References
[1] Golomoziy, V. V., Kartashov, N. V., On coupling moment integrability for time-inhomogeneous Markov chains, Theor. Imov. and
Math. Statistics 89, 2014, 1–12.
Taras Shevchenko National University of Kyiv, Volodymyrska Str. 64, Kyiv 01601, Ukraine
12
GENERALIZATION OF DOOB-MEYER DECOMPOSITION
THEOREM AND ITS APPLICATIONS
N. S. Gonchar
In report we present results concerning optional decomposition of supermartingale relative to convex set of equiv-
alent measures. These results generalize Doob-Meyer decomposition theorem for supermartingale [1, 2].
Theorem 1. Let {ft,Ft}
t=0 be cadlag supermartingale relative to convex set of equivalent measures Mbeing deﬁned on
measurable space {,F} and let there exist a real number 0< t0<,a measure ˜
PMsuch that E˜
Pft0< E ˜
Pft0.
If
sup
PM
EPsup
t[0,d]|ft|=Fd<,0d < ,
then there exist Ft-adapted increasing process g={gt}
t=0, g0= 0,and a right continuous martingale {Mt,Ft}
t=0
relative to convex set of equivalent measures Msuch that
ft=Mtgt.
References
[1] Meyer, P. A., A decomposition theorem for supermartingales. Illinois J. Math. 7, 1963, 1–17.
[2] Meyer, P. A., A decomposition for supermartingales: the uniqueness theorem. Illinois J. Math. 6, 1972, 193–205.
Bogolyubov Institute for Theoretical Physics of NAS of Ukraine, Metrolohicha Str. 14b, Kyiv 03680, Ukraine
STATISTICAL MODEL OF THE STOCK MARKET ORDER QUEUES
A. I. Kakoichenko
The price modeling problem is usually considered as a time series analysis problem. But this approach is not
eﬀective for high frequency trading applications, because it misses the state of the order book and information about
transactional ﬂows.
In recent years, with the growth of electronic exchanges and algorithmic trading, the availability of high frequency
data on the transactional ﬂows opens new opportunities for creating ﬁnancial market models for short-term forecasting
and high frequency trading algorithms development. In this paper we develop a model for markets with continuous
double auction.
Following the seminal work by Avellaneda and Stoikov [1] the ﬂow of transactions of placing new orders is a Poisson
ﬂow. Korolev in his work [2] considers the time spans between two transactions as gamma-distributed. But the analysis
of transactional ﬂows, made in this work, proved, that this assumption contradicts the reality.
The state of the market with continuous double auction can be considered as states of order books of buy and sell
orders respectively. Order book is a queue of orders with equal direction, where new orders with opposite direction
could process these orders or could be placed to queue of orders with their direction. The process of order book
changing is unambiguously described by streams of transactions. There are three most popular types of transactions:
add order, remove order and modify order. The proposed model based on the range of frequencies gives the opportunity
to generate events and modify the initial state to build a prediction of the orders books states. Application of this
model to historical data showed wide prospect of this approach.
References
[1] Avellaneda, M., Stoikov, S., High-frequency trading in a limit order book. Quantitative Finance 8(3), 2008, 217–224.
[2] Korolev, V. Yu., Chertok, A. V., Korchagin, A. Yu., Gorshenin, A. K., Probability and statistical modeling of information ﬂows in
complex ﬁnancial systems based on high-frequency data. Informatics and its Applications 7(1), 2013, 12–21.
Taras Shevchenko National University of Kyiv, Volodymyrska Str. 64, Kyiv 01601, Ukraine
SIMULATION OF FINANCIAL SERIES WITH DESIRED
MULTIFRACTAL PROPERTIES
L. Kirichenko1, A. Khabachova, A. Storozhenko
Lately numerous studies have identiﬁed a number of speciﬁc features of ﬁnancial series: peakedness, heavy-tailed
distributions, property of self-similarity, long-term dependence, conditional volatility et. In 1996, B. Mandelbrot
proposed multifractal model describing the dynamics of ﬁnancial time series MMAR (Multifractal Model of Asset
Returns) [1]. It is based on modeling of fractional Brownian motion in multifractal time by operation of subordination.
In the general case operation of subordination (random substitution of time ) can be represented in the form
Z(t) = Y(T(t)),t > 0, where T(t)is nonnegative nondecreasing stochastic process called subordinator. Y(t)is
stochastic process, which is independent of T(t).
In [1] it is proved that if a stochastic process X(t)is process of subordination X(t) = BH(θ(t)), where BH(t)is
fractional Brownian motion with Hurst parameter Hand θ(t)is subordinators, which is the distribution function of
the cumulative measure deﬁned on the interval [0, T ], then X(t)is the multifractal process with multifractal spectrum
fX(α) = fθ(α
H), where fθ(α)is multifractal spectrum of the process θ(t).
13
In [2] it is suggested the method of building a stochastic binomial cascade with weights having a beta-distribution.
The proposed algorithm allows to generate a realizations of the desired values of the Hurst exponent Hand the
multifractal spectrum f(α). Thus we can simulate the ﬁnancial time series using the cascade realizations with the
required fractal characteristics.
In this work we have investigated various ﬁnancial time series: stock prices, indices, currency pairs, which showed
their obvious multifractal properties. The proposed approach allows to model the realizations with the desired multi-
fractal characteristics.
References
[1] Calvet, L., Fisher, A., Mandelbrot, B., Large deviation and the distribution of price changes. Cowles Foundation Discussion Paper
1165, 1997, 1–28.
[2] Kirichenko, L., Radivilova, T., Kayali, E., Modeling telecommunications traﬃc using the stochastic multifractal cascade process.
Problems of Computer Intellectualization, Kiev-Soﬁa: ITHEA, 2012, 55–63.
Kharkiv National University of Radioelectronics, Lenin Ave. 14, Kharkiv 61166, Ukraine
THE RATE OF CONVERGENCE OF OPTION PRICES
FOR THE DISCRETIZATION OF GEOMETRIC
ORNSTEIN-UHLENBECK PROCESS
Yu. S. Mishura
We take the martingale central limit theorem that was established, together with the rate of convergence, by
Liptser and Shiryaev ([1]), and adapt it to the multiplicative scheme of ﬁnancial markets with discrete time that
converge to the standard Black-Scholes model. To improve the rate of convergence, we suppose that the increments
are independent and identically distributed (but without binomial or similar restrictions on the distribution). Under
additional assumptions, in particular under the assumption that absolutely continuous component of the distribution
is nonzero, we apply asymptotical expansions of distribution function and establish that the rate of convergence is
O(n1/2).The results are contained in [2–4].
References
[1] Liptser, R. S., Shiryaev, A. N. Theory of Martingales. Mathematics and its Applications. Kluwer, Dordrecht, 1989.
[2] Mishura, Yu., Diﬀusion approximation of recurrent schemes for ﬁnancial markets, with application to the Ornstein-Uhlenbeck process.
Opuscula Mathematica 35 (1), 2015, 99–116.
[3] Mishura, Yu., The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black-Scholes
model, in Advances in Mathematics of Finance, ed. A. Palczewski and L. Stettner, BCP 104, 2014.
[4] Mishura, Yu., The rate of convergence of option prices on the asse