TARAS SHEVCHENKO NATIONAL UNIVERSITY OF KIEV INTERNATIONAL CONFERENCE MODERN STOCHASTICS: THEORY AND APPLICATIONS. IV May 24-26, 2018, Kyiv, Ukraine C O N F E R E N C E M A T E R I A L S TARAS SHEVCHENKO NATIONAL UNIVERSITY OF KYIV INTERNATIONAL CONFERENCE MODERN STOCHASTICS: THEORY AND APPLICATIONS. IV Dedicated to the 100-th anniversary of I.I.Gikhman May 24–26, 2018, Kyiv, Ukraine CONFERENCE MATERIALS ORGANIZED BY Taras Shevchenko National University of Kyiv V.M. Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine National Pedagogical Dragomanov University PROGRAM COMMITTEE Yu. Mishura, Yu. Kozachenko, M. Gorodnii, O. Iksanov, P. Knopov, Yu. Kondratiev, V. Koshmanenko, A. Kukush, N. Kuznetsov, E. Lebedev, R. Maiboroda, O. Nakonechnyi, M. Pratsiovytyi, V. Radchenko, L. Sakhno, G. Shevchenko, O. Stanzhitskyi, G. Torbin ORGANIZING COMMITTEE Yu. Mishura (Co-Chair), Yu. Kozachenko (Co-Chair), M. Gorodnii, T. Ianevych, O. Iksanov, P. Knopov, Yu. Kondratiev, N. Kuznetsov, E. Lebedev, R. Maiboroda, O. Nakonechnyi, M. Pratsiovytyi, V. Radchenko, L. Sakhno, G. Shevchenko, O. Stanzhitskyi, G. Torbin, I. Bodnarchuk, O. Borisenko, V. Golomozyi, T. Moklyachuk, O. Ragulina, K. Ralchenko, O. Vasylyk, V. Zubchenko CONTENTS On the 100th anniversary of the birth of I.I. Gikhman . 4 V.S. Koroliuk. I.I. Gikhman, the ally of A.V. Skorokhod . 12 В.С. Королюк. И.И. Гихман – соратник А.В. Скорохода. .13 I.N. Kovalenko. About my dear University teacher Iosif Ilyich Gikhman . 14 Yu.S. Mishura. Influence of ideas of I.I. Gikhman on the worldview of the younger generation of mathematicians . 16 Ю.С. Мiшура. Вплив їдей Й.I. Гiхмана на свiтогляд молодшого поколiння математикiв . 17 L.E. Shaikhet. From the memoirs about Iosif Ilyich Gikhman . 18 Л.Е. Шайхет. Из воспоминаний о Иосифе Ильиче Гихмане. .19 Conference materials . 20 3 On the 100th anniversary of the birth of I.I. Gikhman I.I. Gikhman May 26, 2018 marks the 100th anniversary of the birth of Iosif Ilyich Gikhman, a world-wide known mathematician, expert in the field of probability theory and mathematical statistics, with a great number of important pioneering results which now have become classical ones, and who was one of the founders of the Ukrainian school of probability theory and theory of stochastic processes. Iosif Ilyich Gikhman was born in Uman in the Kiev province (now Cherkasy region of Ukraine) on May 26, 1918. In 1935 he enrolled at, and in 1939 graduated from the Physics and Mathematics Faculty of Kiev University and began his scientific career in postgraduate studies under the supervision of N.N. Bogolyubov. I.I. Gikhman in his younger years Scientific work was interrupted by the World War II, in which I.I. Gikhman had participated (1941 – 1945). Just before the war, he already prepared a Ph.D. (Candidate of Sciences) thesis “Dynamic systems under the influence of random forces”. But he did not have time to defend it before leaving to the front. Throughout the war I.I. Gikhman had been staying in the army, was wounded, and had military awards. During a short leave from the army, in 1942 he defended his thesis at the Institute of Mathematics of the Uzbek Academy of Sciences. After the war, I.I. Gikhman worked first at the Kiev Automobile and Highway Institute, and from 1948 to 1966 he worked at the Kiev State University as an assistant professor, professor, and a head of the Department of Probability Theory. In 1955, he defended his doctoral dissertation “Markov processes and some problems of mathematical statistics”, and in 1959 I.I. Gikhman received the title of professor. Department of Probability Theory and Mathematical Statistics, 1960s Top row: D. Gusak, G. Butsan, R. Boiko, V. Anisimov, D. Silvestrov, A. Turbin Bottom row: I. Ezhov, G. Syta, I. Gikhman, A. Dorohovtsev, A. Skorokhod, M. Yadrenko, M. Portenko, V. Koroliuk, L. Lobanova 4 In 1965, I.I. Gikhman was elected a Corresponding Member of the Academy of Sciences of the Ukrainian SSR. Since 1966 he had been in charge of the Department of Probability Theory of the Institute of Applied Mathematics and Mechanics of the Academy of Sciences of the Ukrainian SSR in Donetsk and directed the scientific work of the Department of Probability Theory at Donetsk State University. Iosif Ilyich Gikhman died on July 30, 1985. I.I. Gikhman’s works are devoted mainly to various problems of mathematical statistics and the theory of random processes. He made the most significant contribution to the theory of stochastic differential equations, being one of the creators of this important research area in probability and stochastic processes. In his first works, developing the ideas of N.N. Bogolyubov, I.I. Gikhman began to study the behavior of dynamical systems under the influence of random perturbations. Generalizing the concept of a dynamical system under the influence of a random process, in 1947 he came to the definition of a stochastic differential equation. In the subsequent papers, the conditions for the existence and uniqueness of solutions of stochastic differential equations were investigated, the dependence on the initial data was studied, and the inverse Kolmogorov equations for the transition probabilities were derived. Also, Bogolyubov’s averaging method was proved for random processes that are solutions of stochastic differential equations, and certain classes of stochastic partial differential equations were studied. The second important direction of I.I. Gikhman’s research is the limit theorems for stochastic processes. Here, he founded several new directions. He proposed a general method for constructing asymptotic expansions for distributions of additive functionals of a Markov chain converging to a continuous diffusion process (1952). He obtained first results about the convergence of functionals of the sequence of sums of independent random variables for the case, where the limiting process with independent increments is discontinuous (1953). I.I. Gikhman first discovered the fact of the lack of invariance in limit theorems for certain classes of additive functionals, such as the number of intersections of a certain domain by a random curve (1957). I.I. Gikhman’s research in mathematical statistics is characterized by the consistent application of methods of the theory of random processes and limit theorems for constructing statistical tests. He studied the influence of the data grouping on the limiting distributions of the Kolmogorov and Smirnov test statistics, and also constructed some new nonparametric tests, which take into account the heterogeneity of the sample space (1952 – 1958). The Kolmogorov- Smirnov tests are often applied to parameter-dependent distributions, and the parameters themselves are estimated through some statistical procedure. I.I. Gikhman investigated how such procedure influences the limiting distribution of the test statistic. It turned out that the limiting distribution depends on the type of the initial distribution essentially (1953 – 1956). In 1970, I.I. Gikhman won an M.M. Krylov prize of the Academy of Science of Ukrainian SSR (together with A.V. Skorokhod), and in 1982, he was awarded the State Prize of the Ukrainian SSR. For his significant contribution to pure and applied research, I.I. Gikhman was awarded the Order of the Badge of Honour. I.I. Gikhman was not only a talented scientist, who started a number of new research areas in probability theory and mathematical statistics, but also a remarkable teacher, educator of students and young scientists. Iosif Ilyich has always been very serious about pedagogical activity. Many generations of students at Kiev University, as well as mathematicians working in different fields, recall with great warmth his remarkable lectures on mathematical analysis, probability theory, integral equations and calculus of variations, functional analysis, number theory. His pedagogical activity played an important role in the formation of Donetsk University. The special courses taught by Iosif Ilyich were devoted to the most topical issues of modern probability theory. Thirty three scientists wrote their Ph.D. (Candidate of Sciences) theses under his supervision: V.V. Baklan, V.M. Bandura, B.V. Bondarev, Ya.S. Brodsky, O.S. Chani, A.Ya. Dorogovtsev, M.O. Diesperova, I.I. Ezhov, A.P. Ga- tun, H. Hekendorf, O.A. Illiashenko, I.I. Kadyrova, M.A. Karabash, V.M. Klepikov, A.M. Kolodii, B.M. Kucher, Li Shi Lin, Yu.M. Linkov, B.Ya. Lukacher, S.Ya. Makhno, Kh. Mansurov, S.A. Melnyk, T.S. Piasetska, Ye.I. Poliakova, M.I. Portenko, L.Ye. Shaikhet, A.D. Shatashvili, A.Yu. Shevlyakov, A.Ye. Skvortsov, O.F. Taraskin, M.Y. Yadrenko, Yu.V. Zhaurov, L.A. Zuev. His books “Introduction to the Theory of Random Processes”, 3 volumes of “The Theory of Stochastic Processes”, “Stochastic Differential Equations”, “Controlled Stochastic Processes”, “Stochastic Differential Equations and Their Applications”, co-authored with A.V. Skorokhod, were translated into English and played an exceptional role in the development of modern probability theory and theory of stochastic processes. Here is the Introduction to the English edition of the “Introduction to the Theory of Random Processes”, 1996: “Although there are many books available on the theory of random processes, there are few books featuring rigorous exposition that are suitable for elementary instruction. This is such a book. Designed for students who have had a general course in probability theory, it covers general topics in the theory of random processes (including measure theory and axiomatization of probability theory) as well as more specialized questions: processes with independent increments, Markov processes and limit theorems for random processes. A significant feature of the book is a unique and beautiful treatment of processes with independent increments, which assumes no prior knowledge of the theory of infinitely divisible laws. A wealth of results, ideas and techniques distinguish this text, one of the first to survey in a rigorous way the more modern results in the theory of stochastic processes and to link them to earlier developments in the subject.” The books by I.I.