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

Anatolii Volodymyrovych Skorokhod was born on September, 10, 1930, in Nikopol, 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 recollections, their children (they had two sons) grew in the atmosphere of the parents' various interests, 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 for himself and became an academician in the field of physics.
The parents worked mostly in different villages and small miner's towns moving occasionally to a new place. In 1935 they settled in the city of Marganets, where in 1937 Anatolii went to school. Schooling was interrupted by the war, and Anatolii had to continue his studies at home.
In 1946, the family, fleeing from the famine in the Dnipro region, temporarily moved to town of Kovel (Volyn region of Western part of Ukraine) forests and coppices of which replaced such fimiliar from childhood steppe, boundless, burnt by sun, with water-melon plantations and corn fields.
The contrast in the tenor of life was striking. There in Volyn folk traditions were kept, and polyphonic choruses, in which very young singers carefully performed their parts side by side with the adults, were especially impressing. Although Anatolii lived here, 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 working as a sea captain, but because of his near 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 admitted to Taras Shevchenko Kyiv State University, Department of Mechanics and Mathematics.
Studies were interesting and came easy to him. His aptitude for research first manifested itself when he was a student. Skorokhod decided to specialize in probability theory at the Department of  Mathematical Analysis. These investigations were carried out under the substantial influence of Prof. B.V.Gnedenko and Prof. I.I.Gikhman (who later became a close friend and colleague of Skorokhod). The young man actively joined the scientific work. He had been working on several problems at the same time. After graduation Skorokhod became the author of five scientific papers, three of them were published in the leading scientific journals “Успехи матем. наук''  [“Soviet Math. Surveys''] and “Докл. Акад. наук СССР'' [“Soviet Math. Dokl.''], the rest two were published in the collection of scientific works of the students 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 (1961). To continue his education Skorokhod went to Moscow and became a postgraduate student of Prof. E.B. Dynkin at the Moscow University (1953-56). It was a period of swift development of investigations in the field of the probability theory at Moscow University when foundations were laid for the theory of random processes. A.N.Kolmogorov gathered around himself a large group of talented young people. Soon the personality of a young scientist from Kyiv began to stand out among them by his profound knowledge and a lot of unexpected ideas. They say that young colleagues had appreciated every possibility (for example, in a long queue in the University cafeteria) to contact Skorokhod and they usually recevied the answers to their questions.
Skorokhod's works of this period were already full of original approaches and unusual associations. It was just then that he proposed the topology in a space of functions without discontinuities of the second kind. This topology served as an instrument for proving limit theorems for the wide class of random processes and now it has the name of Skorokhod topology in the literature. He also created a principally new approach to the proof of the limit theorems (which is now well-known as the method of a single 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 find the final result, the necessary and sufficient conditions of the statements.
In 1957, Skorokhod returned to Kyiv and began his work as a lecturer at the 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, while continuing with his lectures at the 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 the Kyiv University Skorokhod distinguished himself by a unique manner of delivering lectures. His creative imagination often lead him to improvisations, making his students the participants of creative scientific work. When Skorokhod returned to Kyiv, the work of the scientific seminar on probability theory at the Kyiv University became much more 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 problem considered to other problems turned the seminar sessions into a real creative laboratory, and all interested scholars tried to deliver a talk at the seminar in the presence of Skorokhod. Thus, the Kyiv school in probability theory was largely formed as a result of Skorokhod's  activities.
Since the middle 1950s, Skorokhod's works have played a fundamental role in development of the theory of random processes;  to a great extent, they determined the directions of further  investigations in this theory not only in the Ukraine but also in the entire 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 accomplished the series of attempts of numerous mathematicians aimed at the generalization of the famous Donsker  invariance principle to the case where the limit process is an arbitrary, not necessarily continuous, process with independent  increments. In these works, Skorokhod demonstrated his  extraordinary power of independent thought and constructive imagination. In these papers, which formed the basis for his Candidate-Degree thesis, Skorokhod proposed the method of a single probability space (mentioned earlier) and introduced several topologies in the space of functions that do not have discontinuities of the second kind, one of which is now widely known as the Skorokhod topology. These tools enabled him to completely solve all problems related to the aforementioned 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 the problem of choice of a particular group of methods makes sense only with respect to an individual problem; the advantage of analytic methods lies in their universality, whereas the advantage of probability methods is their  close relation to the essence of the problem.
The theory of stochastic differential equations is the most significant branch of probability theory in which direct probability methods are largely used, and it is quite natural that this theory  drew Skorokhod's attention. As a result, he immediately obtained several significant results, which 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 by the 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 in manifolds with boundary. These results aroused much interest all over the world and stimulated numerous deep investigations on the problem of constructing processes of this type. Later, Skorokhod continued the investigation of this problem [see his 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 describing the local structure of any continuous Markov process or, say, process that does 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, one should mention the books “Integration in Hilbert Spaces'', ``Random Linear  Operators'', and “Theory of Random Processes'' (in 3 volumes; written jointly with I.I.Gikhman) – a fundamental monograph  reflecting the contemporary state of the most branches of the theory of random processes. 
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 school in probability theory can hardly be overestimated. He has more than 50 disciples, among which there are 17 Doctors of Sciences. His lectures on all branches of contemporary 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 (and, there are 22 translations of these monographs) and more than 300 works published in scientific journals, he headed numerous scientific seminars, etc.
Skorokhod paid considerable attention to widespread mathematical knowledge. He wrote textbooks and popular-science books (as a whole, he is an author of 16 textbooks and popular-science books), he 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 seventies and eighties. 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. This supported creation of museums or other memorial places in honour of outstanding mathematicians of the past (for example, G. Voronoї, V. Bunyakovskiї, M. Kravchuk).
Skorokhod always was distinguished by his independent opinion. He always stood his ground, though this was quite dangerous under the totalitarian regime. In 1968, he was one of Ukrainian intellectuals who signed the letter 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, advise post-graduates, he was excluded from the Editorial Boards of some scientific journals. And for fifteen years, he was not permitted to participate in scientific conferences abroad. 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 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 1990s 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 together with Habib Salehi and Frank Hoppensteadt (Springer, 2002, Ser. Appl. Math. Sci., № 150).

His scientific interests during the last years of his life were related to the problem of constructing a stochastic process in the space of configurations. Such processes are useful in some modern theories in physics and economics.

The contribution of A.V.Skorokhod into the theory of stochastic processes is universally recognized. His ideas, the methods proposed by him and the results obtained by him will always serve to the development of mathematics. And his path of life, his courage, his decency, his love for Ukraine will remain for his colleagues and disciples as the main life's precept.