2014-12-94

ObituaryA few words about the first book by A. V. SkorokhodI was a fourth-year student at Kiev University when the firstbook by A. V. Skorokhod “Studies in the theory of randomprocesses” was published (1961). By that time, I had alreadytaken the course on the theory of stochastic differential equationstaught by him and that facilitated my efforts in comprehendingthe book. Even so, to read it was an uphill strugglefor me but I didn’t give up. Moreover, I managed to read itwhile serving in the Soviet Army (1964–1965).More than 50 years have passed since then. Many otherbooks on the topic have been published by various authors(including Anatolii Volodymyrovych himself). However,for many researchers of my generation, the first bookby A. V. Skorokhod still remains a spark that has stimulatedtheir enthusiasm for the theory of stochastic processes. Thepower of Skorokhod’s creative ability and the courage of hissearching mind were displayed in that book as brilliantly asfive years before in his fundamental work “Limit theorems forstochastic processes” (1956). I will consider some aspects ofthe book in this article.With more than 50 years experience in reading mathematicalworks by A. V. Skorokhod, I should say that besides theusual difficulties felt by everyone when trying to comprehendsomething new, certain troubles connected with Skorokhod’smanner to expound the material arose. In my opinion, AnatoliiVolodymyrovych was not always irreproachable in thisrespect. One can sometimes come across a sentence in histexts that can be treated in several different ways and it isdifficult (particularly for young mathematicians) to perceivewhich one he intends. I once pointed out this carelessnessthat occasionally existed within his style. He replied that hewas not able to understand why a reader should put into awritten sentence a sense other than the author did. But I wasnot going to give in and said: “It is the author’s responsibilityto structure any phrase in such a way that makes it clear forany reader what the author has meant.” He replied with thequestion: “Do you really believe it possible to read a mathematicaltext without thinking over it?” I then understood hisposition. He showed no concern for particularising the materialexpounded in his works. To think over new problems wasmore important to him than to expound the thoughts and ideasalready discovered by him or others.There is another source of difficulties in reading the textsof A. V. Skorokhod. According to his own confession, if hemade a mistake when writing a sentence or a formula, he didnot notice it when re-reading: instead of what was written hesaw what should be written.In conclusion to this introductory part of the article: Skorokhod’stexts are not straightforward but it is worth it to readthem.***A. V. Skorokhod started lecturing at Kiev University in 1957.Over the three previous years, he was a postgraduate studentat Moscow University. His studies there were a dazzling success:he had formulated and proved the most general limit theoremsfor stochastic processes and, moreover, he had inventedan original method for proving them. In spite of his youngage, he had already succeeded in gaining authority amongstexperts in probability theory. His PhD thesis (candidate dissertation)had already been prepared for defence and rumourswere around that the second scientific degree, i.e. doctorateof mathematics, would be conferred to him for that thesis (inreality it turned out not to be so for a reason that will not bediscussed here).In such a situation, it would be natural for everyone totake a pause in scientific research but not Anatolii Volodymyrovych.His list of publications shows that his searchingmind was working incessantly. However, the main field of hisscientific interests was changing: the theory of stochastic differentialequations started attracting his attention. This theoryhad just been originated by the early 1950s. It would not beright to speak of its details here. I only say that the theory wasindependently created by K. Itô (Japan) and I. I. Gikhman(Kiev) in their works published during the 1940s and early1950s.K. Itô developed the theory of stochastic differential equationsbased on the notion of a stochastic integral he had introduced.His notion was a generalisation of Wiener’s in twodirections: firstly, the integrand in his notion was a randomfunction (Wiener constructed the integral of a non-randomone) and, secondly, he constructed integrals not only withrespect to Brownian motion but also with respect to a (centered)Poisson measure. Given some local characteristics of astochastic process to be constructed and, besides, such “simple”objects as Brownian motion and a Poisson measure, hewrote down a stochastic integral equation (it could be writtenas a differential one) whose solution gave the trajectoriesof the process desired. Under some conditions on given coefficients(the local characteristics mentioned above), he managedto prove the existence and uniqueness of a solution andto establish it as a Markov process. The set of stochastic processesthat were differentiable in Itô’s sense was endowed bya calculus different from the classical type (for example, theItô formula created a new rule of differentiating a functionof a stochastic process having Itô’s stochastic differential).The approach to the theory of stochastic differential equationsgiven by K. Itô turns out to be exceptionally proper: in mostof the monographs on the topics, the basic notion is the notionof 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 rigorousin a mathematical sense. It was based on the notion ofa random field that locally determined the increments of theprocess to be constructed as a solution to the correspondingstochastic differential equation. Under some conditions on agiven random field, I. I. Gikhman proved the theorem on theexistence and uniqueness of a solution with given initial data.If the random field did not possess a property of after-effectthen the solution was a Markov process. In the case of thatfield being given by a vector field of macroscopic velocitiesplus the increments of Brownian motion transformed by agiven operator field, the corresponding solution turned out tobe a differentiable function with respect to the initial data (underthe assumption, of course, that the mentioned fields weregiven by smooth functions in spatial arguments). With thisresult, I. I. Gikhman managed to prove the theorem on the existenceof a solution to Kolmogorov’s backward equation (i.e.a second-order partial differential equation of parabolic type)EMS Newsletter December 2014 27