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Anatolii Skorokhod memory page

A short sketch of the life and research of
Anatolii Volodymyrovych Skorokhod

Anatolii Volodymyrovych Skorokhod was born on September, 10, 1930, in Nikopol. It was an industrial city of Dnipropetrovs'k region in the south of Ukraine. His father, Volodymyr Oleksiyovych, was a teacher of mathematics, physics and astronomy. His mother, Nadiya Andriivna, besides mathematics, taught also history, literature, and music. According to Nadiya Andriivna's memories, their children (they had two sons) grew in the atmosphere of the various interests of their parents, with love to books and nature. The parents treated them with care, respecting their desires and inclinations. Probably, it was not by chance that the younger son, Valerii, who admired his elder brother, also had chosen the scientific career and became an academician in the field of physics.

The parents worked mostly in different villages and small miner's towns moving from time to time to a new place. In 1935 they settled in the city of Marganets, where Anatolii went to school in 1937. Schooling was interrupted by the war, and Anatolii had to continue his studies at home. In 1946, the family, fleeing from the famine raging across the Dnipro region, temporarily moved to Kovel (a town in Volyn region in the Western part of Ukraine). Although Anatolii had lived there, in Kovel, for quite a short time, the nature of this region and especially the national spirit of the locals left a lasting impression on him. At that time, while dreaming about his future, he imagined himself as a sea captain, but because of his short sight the romantic dream has never come true.

In 1948 Anatolii graduated from Kovel's secondary school (received a golden medal as a reward for academic achievements) and was accepted to Taras Shevchenko Kyiv State University, Department of Mechanics and Mathematics. Studies there were interesting and came easy to Skorokhod. He decided to specialize in probability theory at the Department of Mathematical Analysis. His investigations were carried out under the substantial influence by Prof. B.V.Gnedenko and Prof. I.I.Gikhman. The later became a close friend and colleague to Skorokhod. The young man actively joined the scientific work. He had been working on several problems at the same time. He graduated from the University being already an author of five scientific papers, three of them were published in the leading scientific journals "Soviet Math. Surveys" and "S.R. (Dokl.) Acad. Sci. URSS" (Doklady Akademii Nauk SSSR), the rest two were published in the collection of the students scientific works of Kyiv State University. It is worth mentioning that two of these early Skorokhod's works were translated into English and published in “Selected Translations on Mathematical Statistics and Probability” in 1961.

To continue his education after graduation from the University Skorokhod went to Moscow and became a postgraduate student of Prof. E.B. Dynkin at Moscow University (1953-56). It was the heyday for a development of investigations in the field of the probability theory at Moscow University. At that period the foundations for the theory of random processes were laid. A.N.Kolmogorov has attracted a large group of talented young people. Profound knowledge and a lot of unexpected ideas of a young scientist from Kyiv distinguished him among others. Young colleagues had appreciated every possibility to communicate with Skorokhod (when standing, for example, in a long queue in the University cafeteria) and he usually had the answers for their questions.

Skorokhod's works of that period were full of original approaches and unstandard associations. At this time he proposed the concept of the topology in a space of functions without discontinuities of the second order. This topology served as an instrument for proving limit theorems for the wide class of random processes and it was named as Skorokhod topology in the literature. He also created a principally new approach for the proving the limit theorems (which is now well-known as the method of common probability space), a method of pure probabilistic investigation, dealing with random variables instead of their distribution functions. The characteristic feature of Skorokhod's research was his urge to the completeness of the result, to find the necessary and sufficient conditions of the statements.

In 1957, Skorokhod returned to Kyiv and started his work as a lecturer at Kyiv University. In 1964, he became the Head of the Department of the Theory of Random Processes at the Institute of Mathematics of the Ukrainian Academy of Sciences, continuing his work as a lecturer at Kyiv University. For his scientific results, Skorokhod received numerous titles and degrees: Doctor of Sciences, Professor (1963), Corresponding Member of the Ukrainian Academy of Sciences (1967), Academician of the Ukrainian Academy of Sciences (1985), and a Member of the American Academy of Art and Science (2000). In 1982 and in 2003, he was awarded the Ukrainian State Prize in Science and Technology.

From the very beginning of his career at Kyiv University Skorokhod standed out with his unique manner of delivering lectures, proving numerous statements impromptu, engaging students to the creative scientific work. When Skorokhod returned to Kyiv, the work of the scientific seminar on probability theory at the Kyiv University became very active. His discussions with speakers and capability to understand the core of a problem, generalize it, find possible weak points in the proof, and reveal the hidden relation of the considered problem to the other problems made the seminar sessions into a real creative laboratory. All the interested scholars tried to deliver a talk at the Skorokhod seminar. Thus, the Kyiv probabilistic school was largely formed due to Skorokhod's activities.

Since the middle 1950s, Skorokhod's works have played a fundamental role in the development of the theory of random processes. To a great extent, they determined the directions of the further investigations in this theory not only in Ukraine but also in the world. The first series of Skorokhod's works that gained him wide recognition were devoted to the limit theorems for random processes constructed on the basis of sums of independent random variables. These works completed the series of attempts of numerous mathematicians to generalize the famous Donsker invariance principle for the case when the limit process is an arbitrary, not necessarily continuous, process with independent increments. In these works Skorokhod demonstrated his outstanding creative force and original type of thinking. In the papers, which formed the basis for his Candidate-Degree thesis, Skorokhod proposed the method of common probability space and introduced several topologies in the space of functions that do not have discontinuities of the second kind. One of such topologies is now widely known as the Skorokhod topology. These tools enabled him to solve problems related to the mentioned above generalization of the Donsker invariance principle.

As early as in the works indicated, Skorokhod demonstrated his inclination in favor of direct probability methods for solving problems of probability theory. In the preface to his first monograph Studies in the Theory of Random Processes'' (Kyiv University, Kyiv, 1961), he wrote that wrote that the problem of choice of a particular group of methods makes sense only with respect to an individual problem; the advantage of analytical methods is their universality, while the advantage of probabilistic methods is their intrinsic relation to the problem.

The theory of stochastic differential equations is the most significant branch of probability theory that uses direct probability methods widely. And it is quite natural that this theory drew Skorokhod's attention. As a result, he obtained several significant results, which had made him one of the leading experts in this branch of mathematics. Among these results, one should mention his proof of the theorem on existence of solutions of stochastic differential equations using his method of a single probability space under the assumption that the coefficients of these equations are continuous functions (i.e., they may not satisfy the Lipschitz condition).

Another important direction in the theory of stochastic differential equations in which Skorokhod obtained fundamental pioneer results at the beginning of the 1960s is related to the equations that describe processes on manifolds with boundary. These results aroused much interest all over the world and stimulated numerous investigations on the problem of construction the processes of such type. Later Skorokhod continued the investigation of this problem [see monograph “Stochastic Equations for Complex Systems'' (Nauka, Moscow, 1983)]. In this monograph, Skorokhod also considered another problem that had drawn his attention in the second half of the 1960s, namely, the problem of description the local structure of all continuous Markov processes or, say, processes that do not have discontinuities of the second kind. In 1966, he proved that a sufficiently broad class of continuous Markov processes can be reduced to quasidiffusion processes by a random change of the time variable. In the monograph “Stochastic Equations for Complex Systems'' he constructed stochastic differential equations for quasidiffusion processes taking values in spaces of complex structure (e.g., manifolds with boundary, manifolds with variable dimensionality, etc.).

Among the works of Skorokhod published in the 1970s, it is worth to mention the books “Integration in Hilbert Spaces”, “Random Linear Operators”' and “Theory of Random Processes” (in 3 volumes; written together with I.I.Gikhman). The last one is a fundamental monograph reflecting the state of the main branches of the theory of random processes at that time. In the 1970s, Skorokhod introduced several notions, which are now widely used not only by mathematicians, but also by physicists. Among them, one should mention the notions of extended stochastic integral (the Skorokhod integral), strong (weak) random linear operator, and stochastic semigroup. The notion of strong random linear operator was used by Skorokhod for the description of the structure of certain classes of stochastic semigroups. These results were published in the monographs “Processes with Independent Increments” (2nd edition, Nauka, Moscow, 1986) and “Asymptotic Methods of the Theory of Stochastic Differential Equations” (Naukova Dumka, Kyiv, 1987). In the latter monograph, Skorokhod applied the notion of stochastic semigroup to the problem of stability of stochastic systems.

Skorokhod's contribution to the formation of the Ukrainian probabilistic school can hardly be overestimated. He had more than 50 disciples, among which there are 17 Doctors of Sciences. His lectures on all branches of theory of random processes presented at the Kyiv University and numerous popular-science works contributed much to the mathematical education of youth. Skorokhod is the author of 23 scientific monographs (most of them were immediately translated and published abroad) and more than 300 works published in scientific journals, he headed numerous scientific seminars, etc.

Skorokhod paid considerable attention to the spread of mathematical knowledge. He wrote textbooks and popular-science books (as a whole, he is an author of 16 textbooks and popular-science books) and delivered lectures on television for students. Every September, he lectured schoolchildren at the opening ceremony of a new school-year of the University for Young Mathematicians which worked at the Institute of Mathematics in the 70s and 80s. With deep understanding, he supported popularizing the names of distinguished Ukrainian mathematicians of the past. For this goal, he undertook several travels to lecture all over Ukraine.

Skorokhod always differed by his independent opinion. He stood on his ground, though this was quite dangerous under the totalitarian regime. In 1968, he took part in the campaign of the group of Ukrainian intellectuals defending the constitutional rights of citizens of the country. All the participants of this group were punished. As a result, Skorokhod was not allowed to lecture students, advice post-graduates. He was excluded from the editorial boards of some scientific journals, and for fifteen years, he had not been permitted to participate in scientific conferences abroad. Skorokhod stood this forced limitation of his rights with proper pride. At that time he told, that mathematics saved him from all the life troubles. During the period 1969-1982 he had worked fruitfully and there had been published 12 his books, among which the three-volume monograph "The theory of stochastic processes" (in collaboration with I.I. Gikhman), and 12 scientific popular books (partly in collaboration). His absence at international scientific conferences gave birth to the opinion among foreign scientists that “Skorokhod” was the collective name of a group of Soviet scientists, just as the group of French mathematicians united under the name “Bourbaki”.

Since 1993, Skorokhod had worked at the Michigan State University (Lansing, Michigan, USA), retaining close scientific relations with the Institute of Mathematics of the Ukrainian Academy of Sciences. His scientific works of these last years were devoted to the investigation of the asymptotic behavior of dynamical systems under random perturbations. Results of these investigations were published in the book "Random perturbation methods with applications in science and engineering", written jointly with Habib Salehi and Frank Hoppensteadt (2002). Also at that time he writes, together with Shlomo Leventhal, a highly reputed article in financial mathematics, "On the possibility of hedging options in the presence of transaction costs". Since 2000, Skorokhod had been a member of the American Academy of Art and Science.

Anatolii Volodymyrovych Skorokhod has passed away on January, 3, 2011 in Lansing, Michigan (USA) and on 20th of May 2011 Anatolii Volodymyrovych Skorokhod ashes were buried at Baikove cemetery in Kyiv.

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