2014-12-94

ObituaryAnatolii Skorokhod(10 Sept 1930 – 3 Jan 2011)(10 September 1930 – 3 January 2011)Valerii Buldygin, Andrey Dorogovtsev, Mykola Portenko (National Academy of Sciences of Ukraine, Kiev, Ukraine)Valerii and Irina Buldygin, Kadyrova Andrey (Michigan Dorogovtsev, State University, Mykola East Portenko Lansing, (National USA) Academy of Sciences of Ukraine, Kiev, Ukraine)and Irina Kadyrova (Michigan State University, East Lansing, USA)Anatolii SkorokhodThis article is devoted to an outstanding mathematician andexcellent teacher A. V. Skorokhod, who recently passed away.The introductory part has been written by V. Buldygin (whodied in 2012), A. Dorogovtsev and M. Portenko. A version ofSkorokhod’s biography is presented by I. Kadyrova, who washis wife from 1975 up to his death. M. Portenko describes hisown impressions about the first book by A. V. Skorokhod, andA. Dorogovtsev presents his point of view on the evolution ofthe notion of the Skorokhod integral and related topics.The name A. V. Skorokhod belongs with the few outstandingmathematicians of the second half of the last centurywhose efforts have imparted modern features to mathematics.His extraordinarily creative potential can be appraised byeveryone who has studied contemporary stochastic analysisand realised that a considerable proportion of its notions andmethods 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 themethod of a single probability space, strong and weak linearrandom operators, and stochastic semigroups (also proposedby him) and the power of his creative capacity becomes clear.Some of these notions (e.g. Skorokhod’s integral) are nowuseful not only in mathematics but also in modern theoreticalphysics.Graduating from the University of Kiev in 1953, A.V. Skorokhod carried on his postgraduate studies at MoscowUniversity from 1953 to 1956, where he had the opportunityto learn from the achievements of the famous Moscowprobabilistic school with academician A. N. Kolmogorov atits head. During this time A. V. Skorokhod gained authorityamong the scientific world when he had managed to formulateand prove the general invariance principle. A particularcase of that principle was known as a result by M. Donsker(established in 1951). However, Skorokhod’s result was nota simple generalisation of M. Donsker’s. In order to formulateand prove it, A. V. Skorokhod introduced several newtopologies into the space of functions without discontinuitiesof the second kind (one of those topologies is now well knownas Skorokhod’s topology and is useful in many branches ofmathematics). Moreover, he proposed an original approach tothe problem on the convergence of probability distributions(the method of a single probability space). Making use ofthose new tools, A. V. Skorokhod formulated and proved thegeneral invariance principle in an accomplished form. Thoseresults are now included in any fundamental monograph onthe theory of stochastic processes. The probabilists of thosedays were deeply impressed by Skorokhod’s new ideas and,in 1956, the most authoritative probabilist A. N. Kolmogorovpublished the paper “On Skorokhod’s convergence”, where hegave his own interpretation of the notions just introduced byA. V. Skorokhod. In one of his papers published in 2000, ProfessorV. Varadarajan of the University of California wrote:I was a graduate student in Probability theory at the IndianStatistical Institute, Calcutta, India in 1956, and still remembervividly the surprise and excitement of myself and my fellowstudents when the first papers on the subject by Skorokhodhimself and Kolmogorov appeared. It was clear from the beginningthat the space D with its Skorokhod topology would playa fundamental role in all problems where limit theorems involvingstochastic processes whose paths are not continuous (but areallowed to have only discontinuities of the first kind) were involved.However, not only the famous Moscow School of ProbabilityTheory had an influence on the scientific work of A. V. Skorokhod.A significant part of his research was devoted tothe theory of stochastic differential equations, originated byI. I. Gikhman, Kiev, in his works in the 1940s–1950s (independently,that theory arose in the works of K. Itô, Japan, atabout the same time). With the influence of I. I. Gikhman,A. V. Skorokhod engaged in scientific investigations in that24 EMS Newsletter December 2014