This article is devoted to an outstanding mathematician and excellent teacher A. V. Skorokhod, who recently passed away. The introductory part has been written by V. Buldygin (who died in 2012), A. Dorogovtsev and M. Portenko. A version of Skorokhod’s biography is presented by I. Kadyrova, who was his wife from 1975 up to his death. M. Portenko describes his own impressions about the first book by A. V. Skorokhod, and A. Dorogovtsev presents his point of view on the evolution of the notion of the Skorokhod integral and related topics.
The name A. V. Skorokhod belongs with the few outstanding mathematicians of the second half of the last century whose efforts have imparted modern features to mathematics. His extraordinarily creative potential can be appraised by everyone who has studied contemporary stochastic analysis and realised that a considerable proportion of its notions and methods were introduced into mathematics by A. V. Skorokhod. It suffices to mention the notions of Skorokhod’s topology, Skorokhod’s space, Skorokhod’s embedding problem, Skorokhod’s reflecting problem, Skorokhod’s integral and the method of a single probability space, strong and weak linear random operators, and stochastic semigroups (also proposed by him) and the power of his creative capacity becomes clear. Some of these notions (e.g. Skorokhod’s integral) are now useful not only in mathematics but also in modern theoretical physics
Graduating from the University of Kiev in 1953, A. V. Skorokhod carried on his postgraduate studies at Moscow University from 1953 to 1956, where he had the opportunity to learn from the achievements of the famous Moscow probabilistic school with academician A. N. Kolmogorov at its head. During this time A. V. Skorokhod gained authority among the scientific world when he had managed to formulate and prove the general invariance principle. A particular case of that principle was known as a result by M. Donsker (established in 1951). However, Skorokhod’s result was not a simple generalisation of M. Donsker’s. In order to formulate and prove it, A. V. Skorokhod introduced several new topologies into the space of functions without discontinuities of the second kind (one of those topologies is now well known as Skorokhod’s topology and is useful in many branches of mathematics). Moreover, he proposed an original approach to the problem on the convergence of probability distributions (the method of a single probability space). Making use of those new tools, A. V. Skorokhod formulated and proved the general invariance principle in an accomplished form. Those results are now included in any fundamental monograph on the theory of stochastic processes. The probabilists of those days were deeply impressed by Skorokhod’s new ideas and, in 1956, the most authoritative probabilist A. N. Kolmogorov published the paper “On Skorokhod’s convergence”, where he gave his own interpretation of the notions just introduced by A. V. Skorokhod. In one of his papers published in 2000, Professor V. Varadarajan of the University of California wrote:
I was a graduate student in Probability theory at the Indian Statistical Institute, Calcutta, India in 1956, and still remember vividly the surprise and excitement of myself and my fellow students when the first papers on the subject by Skorokhod himself and Kolmogorov appeared. It was clear from the beginning that the space D with its Skorokhod topology would play a fundamental role in all problems where limit theorems involving stochastic processes whose paths are not continuous (but are allowed to have only discontinuities of the first kind) were involved.
However, not only the famous Moscow School of Probability Theory had an influence on the scientific work of A. V. Skorokhod. A significant part of his research was devoted to the theory of stochastic differential equations, originated by I. I. Gikhman, Kiev, in his works in the 1940s–1950s (independently, that theory arose in the works of K. Itô, Japan, at about the same time). With the influence of I. I. Gikhman, A. V. Skorokhod engaged in scientific investigations in that theory after coming back to Kiev in 1957. The results of those investigations obtained by him over 1957–1961 formed the basis of his doctoral dissertation and his first book “Studies in the theory of random processes”, published by Kiev University in 1961. The assertions expounded in that book, as well as the methods used by A. V. Skorokhod for proving them, were fundamentally different from those that were typical in the theory of stochastic differential equations at that time: the book was full of new ideas, new methods and new results. At the beginning of the 1960s, A. V. Skorokhod published several articles devoted to the theory of stochastic differential equations that described diffusion processes in a region with a boundary. Those were pioneering works and they stimulated a real stream of investigations on the topic at many probabilistic centres around the world. It should be said that the theory of stochastic differential equations has now become one of the most essential acquisitions of the whole of mathematics in the second half of the 20th century and it is impossible to overestimate the contribution of A. V. Skorokhod.
The full list of Skorokhod’s publications consists of more than 300 articles published in various journals, and 23 monographs, some of them written jointly with co-authors (the number of monographs should be increased to 45 if translations are taken into account). Under Skorokhod’s supervision, more than 50 graduate students defended their candidate dissertations and 17 of his disciples became doctors of mathematics. It should be added that A. V. Skorokhod paid considerable attention to popularising mathematics amongst schoolchildren. He was Rector of the University of Young Mathematicians which was active for 10 years at the Institute of Mathematics in Kiev. Each academic year at that university started with a lecture delivered by A. V. Skorokhod. He published 16 textbooks and popular-science books (some of them with co-authors). A. V. Skorokhod was incessantly in search of new mathematical truth. He was able to see the gist of a problem, to find out an original unexpected approach to it and to create an adequate method for solving it. Besides, he was in the habit of thinking over problems thoroughly every day. Owing to his intense work day after day, the creative spark given to him from God became a bright shining star of the first magnitude on the mathematical frontier.
A brief biographical outline
Anatoli Vladimirovich Skorokhod was born 10 September 1930 in the town of Nikopol, Dnipropetrovsk region (previously Ekaterinoslavskaya province) to a family of teachers. Anatoli spent his childhood in Southern Ukraine. His parents taught in rural schools around Nikopol. Anatoli’s childhood took place during the very difficult 1930s: the ruin after the Revolution and the Civil War of 1919–1922, collectivisation of peasants, dispossession, exile and hunger.
Anatoli’s father Vladimir Alexseevich taught mathematics, physics and astronomy, primarily in high school. A great teacher, he was erudite and had a sharp analytical mind. From him, Anatoli inherited an inquisitive, analytical mind and a critical attitude toward everything. His father played a major role in the choice of his eldest son’s profession. Anatoli’s mother Nadezhda Andreevna taught Russian and Ukrainian literature, history, music and singing, as well as mathematics. Nadezhda Andreevna had many different talents. She was a good musician and had a vivid dramatic talent. Nadezhda Andreevna also had good writing skills. Boasting an excellent style, she wrote scripts, stories and poems.
Anatoli entered elementary school at the age of seven. His studies were interrupted by World War II. The part of Ukraine where the Skorokhods were living was occupied at the beginning of the war.
The post-war years in Southern Ukraine were years of poor harvest and, in 1946, trying to escape from the hunger, the family moved to live in Kovel, a town at Volyn in the western region of Ukraine. Their father was offered a position of school principal. Studying in high school was easy for Anatoli without any apparent effort. He was excellent in all subjects. Despite always being interested in mathematics, during his school years Anatoli did not feel any predestination to become a mathematician.
After his graduation with a gold medal from high school in 1948, Anatoli followed the advice of his father and submitted his documents to the Kiev State University (named after Taras Shevchenko) and was enrolled in the Faculty of Mechanics and Mathematics.
Skorokhod’s scientific work began in his student years. Under the supervision of Boris Vladimirovich Gnedenko (at that time Chairman of the Department of Probability Theory) and Iosif Illich Gikhman (then an associate professor of the department), Anatoli started his work in probability theory. At the end of his student years Skorokhod became involved in the research related to the famous Donsker invariance principle.
During 1953–1956, Anatoli was studying in the graduate school of Moscow State University under the supervision of Eugene Borisovich Dynkin.
This period of study in this graduate school was a remarkable period in Skorokhod’s life in many ways. At this time (the 1950s) in the Faculty of Mechanics and Mathematics, a broad audience of talented young people gathered around the great teachers of the older generation. These young mathematicians saw their future in the service of fundamental science. Amongst this group, Anatoli Skorokhod was distinguished by his independence in research work and the courage and originality of his approaches to problem solving. According to Anatoli, the main thing that he benefited from in Moscow graduate school was the seminar of his advisor E. B. Dynkin, called “Analysis, Algebra and Probability Theory”.
Skorokhod’s PhD thesis (his dissertation was defended in May 1957) contained descriptions of new topologies in the space of functions without discontinuities of the second kind and the application of them for proving limit theorems for stochastic processes. The Donsker invariance principle was generalised to the case when the limit process is a general process with independent increments. In the proofs of the theorems he used the original method invented by the author, known as the “method of a single probability space”. The importance of the ideas of a very young mathematician was confirmed by the entire future development of the theory of stochastic processes. The terms “Skorokhod topology”, “Skorokhod space” and “Skorokhod metric” are included in all basic books on the theory of stochastic processes.
In 1957–1964, Skorokhod was working as a faculty member in his “alma mater” – Kiev University. In 1961, he published his first book called “Studies in the Theory of Random Processes”, which was the basis of his doctoral dissertation, defended in 1963. At the beginning of 1964, at the Institute of Mathematics of the Academy of Sciences of Ukraine, the Department of the Theory of Stochastic Processes was opened and A. V. Skorokhod became head of this department. In the same year he was awarded the title of professor. In 1967, A. V. Skorokhod was elected a corresponding member of the Ukrainian Academy of Sciences.
After Skorokhod’s return from Moscow in 1957, he began a friendship, scientific cooperation and long-term and fruitful co-authorship with I. I. Gikhman. They wrote many well known books together.
A. V. Skorohod played a prominent role in the development of Ukrainian probability theory, particularly in the Kiev school. The scale and diversity of his research and teaching activities were striking. Generations of students grew up listening to his lectures and using textbooks and monographs authored or co-authored by him.
Under A. V. Skorokhod’s leadership (since 1966) the national seminar on probability theory at Kiev State University has gained credibility and relevance not only in Kiev but also far beyond. A. V. Skorokhod supervised graduate students at the university, as well as at the Institute of Mathematics. He was the advisor of 56 PhD students. Among his graduate students were not only Ukrainian students but also young scientists from India, China, Vietnam, East Germany, Hungary, Nicaragua and other countries. Under his guidance, 17 doctoral theses were also written.
A. V. Skorokhod made a great impact helping raise the level of elementary mathematics teaching in Ukraine and the popularisation of mathematics.
In 1993–2011, Skorokhod worked at the Department of Statistics and Probability of Michigan State University, USA. His research areas were the investigation of the behaviour of dynamic systems under random perturbations, some problems of financial mathematics and martingale theory. In 2000, Skorokhod was elected a member of the American Academy of Arts and Sciences.
In an interview, Skorokhod was asked how he felt about social activities of scientists. To that Anatoli replied: “Negatively. I believe that a scientist should be a professional.” However, under circumstances that required the demonstration of personal courage and providing support by his authority in defence of the civil rights and freedoms of citizens, he joined the protesters. In April of 1968, a group of 139 scientists, writers and artists, workers and students wrote a letter to the leaders of the former USSR expressing their concern regarding the renewed practice of closed political trials of young people from the midst of the artistic and scientific intelligentsia. Participation in this event was natural for A. V. Skorohod – a man with a sense of dignity, courageousness and independence, who was never afraid of authority and could not be indifferent to the flagrant flouting of civil rights in the country.
A V. Skorokhod was a true patriot of Ukraine. He hated everything that was part of the concept of “imperial thinking” with regard to Ukraine: denial of identity of language and culture and of the very existence of the distinctive Ukrainian nation, and the rejection of the idea of an independent Ukrainian state. That love for Ukraine made him an active participant of the national liberation movement “People’s Movement of Ukraine” (“Rukh”, late 1980s). Anatoli took part in all activities carried out by the initiative group when the movement was still only emerging. At that time active participation in the creation of “Rukh” was dangerous but that did not stop Skorokhod. When independence of Ukraine was proclaimed and the “Rukh” began to turn into a political, bureaucratic organisation, Skorokhod completely lost interest in it and any participation in its activities.
As a mathematics phenomenon, the extent and significance of “Skorokhod” was due not only to the mathematical talent Anatoli possessed but also an equivalent gift of personality. His mathematical talent, intuition and efficiency caused surprise and delight; his modesty, indifference for awards and titles, an absence of vanity, the independence of his judgments and his inner freedom served as moral standards in his social circle. Skorokhod’s heartwarming subtlety and depth attracted people to him. Erudite in various fields of knowledge, including history and philosophy, he had a love of literature and classical music and a passion for poetry (Skorokhod knew by heart whole volumes of poems of his favourite poets: Ivan Bunin, Osip Mandelshtam, Anna Akhmatova, Boris Pasternak and Joseph Brodsky, and was able to recite them for hours). He was an inspiration for others to follow his example, involving his friends and disciples in the same areas of spiritual and aesthetic interests. Anatoli was a caring, loving son, a loyal, understanding father and a good friend, always ready to support in difficult circumstances, to listen and to help. In personal relations he was very sincere, very romantic and able to love selflessly.
In one of his articles, Anatoli wrote:
Only a curious to oblivion person can be a good mathematician... With the help of mathematics new surprising and unexpected facts are often discovered. In fine art the beautiful creation always contains something unexpected, though not all unexpected is beautiful. Whereas in mathematics unexpected is always beautiful... there is nothing more beautiful than a simple and clear proof of a non-trivial statement.
The engagement in mathematics was for Skorokhod a way of existence as natural as breathing.
“I think about mathematics always,” Skorokhod wrote in one of his letters. The hum of problems he thought about was continuous and incessant in his mind. In his work on problems Skorokhod did not dig deeply in the literature in the search of suitable tools that could be adapted or modified to suit his needs. He created his own original methods and constructions that determined new directions in the development of the theory of stochastic processes for decades. Until the very end of his creative life, Skorokhod maintained an inquisitive curiosity, always searching for harmony and beauty of mathematics.
Anatoli Vladimirovich Skorokhod died in Lansing, Michigan, 3 January 2011. Relatives and friends made a last farewell to Anatoli with the words: “A bright star has returned to the Universe.” The ashes of Anatoli Skorokhod were buried 20 May 2011 at Baikove cemetery in Kiev.
A few words about the first book by A. V. Skorokhod
I was a fourth-year student at Kiev University when the first book by A. V. Skorokhod “Studies in the theory of random processes” was published (1961). By that time, I had already taken the course on the theory of stochastic differential equations taught by him and that facilitated my efforts in comprehending the book. Even so, to read it was an uphill struggle for me but I didn’t give up. Moreover, I managed to read it while serving in the Soviet Army (1964–1965).
More than 50 years have passed since then. Many other books on the topic have been published by various authors (including Anatolii Volodymyrovych himself). However, for many researchers of my generation, the first book by A. V. Skorokhod still remains a spark that has stimulated their enthusiasm for the theory of stochastic processes. The power of Skorokhod’s creative ability and the courage of his searching mind were displayed in that book as brilliantly as five years before in his fundamental work “Limit theorems for stochastic processes” (1956). I will consider some aspects of the book in this article.
With more than 50 years experience in reading mathematical works by A. V. Skorokhod, I should say that besides the usual difficulties felt by everyone when trying to comprehend something new, certain troubles connected with Skorokhod’s manner to expound the material arose. In my opinion, Anatolii Volodymyrovych was not always irreproachable in this respect. One can sometimes come across a sentence in his texts that can be treated in several different ways and it is difficult (particularly for young mathematicians) to perceive which one he intends. I once pointed out this carelessness that occasionally existed within his style. He replied that he was not able to understand why a reader should put into a written sentence a sense other than the author did. But I was not going to give in and said: “It is the author’s responsibility to structure any phrase in such a way that makes it clear for any reader what the author has meant.” He replied with the question: “Do you really believe it possible to read a mathematical text without thinking over it?” I then understood his position. He showed no concern for particularising the material expounded in his works. To think over new problems was more important to him than to expound the thoughts and ideas already discovered by him or others.
There is another source of difficulties in reading the texts of A. V. Skorokhod. According to his own confession, if he made a mistake when writing a sentence or a formula, he did not notice it when re-reading: instead of what was written he saw what should be written.
In conclusion to this introductory part of the article: Skorokhod’ texts are not straightforward but it is worth it to read
them.
* * *
A. V. Skorokhod started lecturing at Kiev University in 1957. Over the three previous years, he was a postgraduate student at Moscow University. His studies there were a dazzling success: he had formulated and proved the most general limit theorems for stochastic processes and, moreover, he had invented an original method for proving them. In spite of his young age, he had already succeeded in gaining authority amongst experts in probability theory. His PhD thesis (candidate dissertation) had already been prepared for defence and rumours were around that the second scientific degree, i.e. doctorate of mathematics, would be conferred to him for that thesis (in reality it turned out not to be so for a reason that will not be discussed here).
In such a situation, it would be natural for everyone to take a pause in scientific research but not Anatolii Volodymyrovych. His list of publications shows that his searching mind was working incessantly. However, the main field of his scientific interests was changing: the theory of stochastic differential equations started attracting his attention. This theory had just been originated by the early 1950s. It would not be right to speak of its details here. I only say that the theory was independently created by K. Itô (Japan) and I. I. Gikhman (Kiev) in their works published during the 1940s and early 1950s.
K. Itô developed the theory of stochastic differential equations based on the notion of a stochastic integral he had introduced. His notion was a generalisation of Wiener’s in two directions: firstly, the integrand in his notion was a random function (Wiener constructed the integral of a non-random one) and, secondly, he constructed integrals not only with respect to Brownian motion but also with respect to a (centered) Poisson measure. Given some local characteristics of a stochastic process to be constructed and, besides, such “simple” objects as Brownian motion and a Poisson measure, he wrote down a stochastic integral equation (it could be written as a differential one) whose solution gave the trajectories of the process desired. Under some conditions on given coefficients (the local characteristics mentioned above), he managed to prove the existence and uniqueness of a solution and to establish it as a Markov process. The set of stochastic processes that were differentiable in Itô’s sense was endowed by a calculus different from the classical type (for example, the Itô formula created a new rule of differentiating a function of a stochastic process having Itô’s stochastic differential). The approach to the theory of stochastic differential equations given by K. Itô turns out to be exceptionally proper: in most of the monographs on the topics, the basic notion is the notion of Itô’s integral or some of its generalisations.
I. I. Gikhman did not have such a notion. Nevertheless, his notion of a stochastic differential equation was quite rigorous in a mathematical sense. It was based on the notion of a random field that locally determined the increments of the process to be constructed as a solution to the corresponding stochastic differential equation. Under some conditions on a given random field, I. I. Gikhman proved the theorem on the existence and uniqueness of a solution with given initial data. If the random field did not possess a property of after-effect then the solution was a Markov process. In the case of that field being given by a vector field of macroscopic velocities plus the increments of Brownian motion transformed by a given operator field, the corresponding solution turned out to be a differentiable function with respect to the initial data (under the assumption, of course, that the mentioned fields were given by smooth functions in spatial arguments). With this result, I. I. Gikhman managed to prove the theorem on the existence of a solution to Kolmogorov’s backward equation (i.e. a second-order partial differential equation of parabolic type) without any assumptions on the non-degeneracy property of the matrix consisting of the coefficients of the second spatial derivatives (it is well known how important such a property is in the analytical theory of those equations). That was a sig nificant result showing that some theorems in the theory of partial differential equations can be proved with the use of purely probabilistic methods.
I have just described briefly the situation in the theory of stochastic differential equations formed by the beginning of 1957. I think that A. V. Skorokhod had the opportunity to acquaint himself with Itô theory during his studies in Moscow. As far as I know, just after coming back to Kiev, his regular discussions with I. I. Gikhman started taking place and he was able to comprehend Gikhman’s approach to the theory. That branch of mathematics was then quite new and A. V. Skorokhod was entirely engaged in investigations on the topic. His works published 1957–1961 (though not all of them) were related to the theory of stochastic differential equations. At the end of 1961 in the publishing house of Kiev University, the first book by A. V. Skorokhod came out. Besides the title, there was a subtitle on its cover, namely “Stochastic differential equations and limit theorems for Markov processes”.
A. V. Skorokhod started the book by setting out Itô theory of stochastic differential equations (more precisely, its multidimensional version) and then he presented several of his new results that essentially influenced the further evolution of the theory.
First, he proved the theorem on the existence of a solution to a stochastic differential equation under the assumption that its coefficients were only continuous (they were also assumed to satisfy the usual growth conditions at infinity, of course), i.e. they might not satisfy the Lipschitz condition in spatial arguments, as was the case in Itô’s or Gikhman’s theories. That theorem was an analogue to the Peano theorem in the theory of ordinary differential equations. The uniqueness of a solution was not guaranteed. It should also be said that the solution constructed by A. V. Skorokhod in proving that theorem seemed to be of somewhat different character from those constructed by K. Itô or I. I. Gikhman. They constructed the solution on the probability space where an initial position, Brownian motion and a Poisson measure (or Gikhman’s random field) were given and that solution turned out to be a functional of those objects. Solutions with this property were later called strong solutions. And A. V. Skorokhod made use of the method of a single probability space (previously invented by him) for proving his theorem and it could not be guaranteed that his solution was a strong one. An absorbing problem then arose: what conditions on given coefficients of a stochastic differential equation one should impose in order to assert that the solution of that equation was strong. A detailed investigation of that problem can be found in the fundamental article by A. K. Zvonkin and N. V. Krylov “On strong solutions of stochastic differential equations” (1974).
Second, for stochastic differential equations describing the processes with jumps, he established the differentiability of a solution with respect to the initial data. This allowed him to derive an integro-differential equation for the corresponding mathematical expectation. That equation was an analogue to the Kolmogorov backward equation for diffusion processes obtained before by I. I. Gikhman (see above).
Third, he found out the conditions under which a pair of stochastic differential equations generated two measures on the space of all functions without discontinuities of the second kind such that one of those measures was absolutely continuous with respect to the other one, and the formula for the corresponding density was written. Those formulae are very important in mathematical statistics when some unknown parameters in the coefficients are to be estimated or one of the two given competitive hypotheses about the coefficients is to be chosen.
Fourth, he formulated and proved a very interesting theorem on comparison of solutions to a given pair of stochastic differential equations on a real line whose diffusion coefficients coincided and whose drift coefficients were related with an inequality valid for all instants of time and at any point of the real line (a term containing a Poisson measure was absent in the equations under those considerations). Then it turned out that the solutions were related with the same inequality as ever, if only their initial positions did so. Making use of this theorem, A. V. Skorokhod established the uniqueness of a solution to a one-dimensional stochastic differential equation with continuous coefficients satisfying the usual growth conditions at infinity and such that the diffusion coefficient was given by a Hoelder continuous function (in spatial argument) with exponent greater than 1/2. As a matter of fact, it was the first result on the existence of a strong solution to an equation with non-Lipschitzean coefficients.
Fifth, he found out the conditions on a given sequence of Markov chains such that the stochastic processes generated by those chains were weakly convergent (in the first of the topologies introduced by him into the space of all functions without discontinuities of the second kind) to the solution of a stochastic differential equation. Some results of the kind had been known by that time for the case of a limiting process being a diffusion one.
Sixth, the last chapter of the book was devoted to the so-called “embedding problem” that could be formulated as follows: for a given one-dimensional Brownian motion, an integrable stopping time was to be constructed such that the value of the Brownian motion at that stopping time was distributed according to a beforehand given measure on a real line being centered and having finite second moment. A. V. Skorokhod solved this problem and applied it to estimating the probability that a sequence of the normalised sums of independent random variables was located inside the region bounded by two given curves. The embedding problem formulated and solved by A. V. Skorokhod in 1960 has stimulated investigations in probability theory for over 50 years. At an international conference “Skorokhod space: 50 years on” that took place in Kiev in 2007, one of its sections had the title “The Skorokhod Embedding Problem”. The organiser of that section, Professor J. Obloj from the UK, published an excellent brief review of the talks on the topic: “The Skorokhod embedding problem: old and new challenges” (see Abstracts of that conference, part 1, pp. 93–97).
The five points above were Skorokhod’s first steps in the theory of stochastic differential equations. It should be mentioned that there was one more step in that theory made in the years 1957–1961, namely his pioneering results related to the theory of stochastic differential equations describing diffusion processes in a region with boundary points. Those results were not included in the book. They originated the theory of stochastic processes now called “The Skorokhod Reflection Problem”. A section with this title, organised by Professors P. Dupuis and K. Ramanan from the USA, also took place at the conference mentioned above.
Skorokhod’s achievements described in this article show how extremely long his first steps were into the theory of stochastic differential equations and how quickly he moved from the position of beginner to one of leader. That theory became his favourite field of mathematics and he had many opportunities after 1961 to think over some of its problems again and again.
Many people knew how great he was, not only as a mathematician but also as a human being. I would like to address anyone who was close to him with the following lines:
Don’t say wistfully: “He is no more . . . ”
But say thankfully: “He was!”
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The remaining part of this article can be found at the website of EMS Letters (see December 2014 Issue, p. 24).