TARAS SHEVCHENKO NATIONAL UNIVERSITY OF KYIV
INTERNATIONAL CONFERENCE
MODERN STOCHASTICS:
THEORY AND APPLICATIONS V
June 1–4, 2021, Kyiv, Ukraine
CONFERENCE MATERIALS
ORGANIZED BY
Taras Shevchenko National University of Kyiv National Pedagogical Dragomanov University
Institute of Mathematics of the National Academy of Sciences of Ukraine
V. M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine Kyiv Mathematical Society
PROGRAM COMMITTEE Yu. Mishura (Chair),
H.-J. Engelbert, M. Grothaus, A. Iksanov, P. Imkeller, A. Ivanov, Yu. Kondratiev, V. Koshmanenko, K. Kubilius,
A. Kukush, N. Kuznetsov, E. Lebedev, N. Leonenko, N. Limnios, R. Maiboroda, D. Marinucci, E. Orsingher,
M. Podolskij, M. R´asonyi, I. Samoilenko, M. Savchuk, R. Schilling, L. Stettner, A. Veretennikov, L. Viitasaari
ORGANIZING COMMITTEE L. Sakhno (Co-Chair), G. Torbin (Co-Chair),
O. Bezushchak, I. Bodnarchuk, O. Borysenko, O. Chernova, V. Golomoziy, O. Hopkalo, T. Ianevych, S. Lohvinenko, S. Maksymenko, T. Moklyachuk, R. Nikiforov, M. Pratsiovytyj, O. Ragulina, K. Ralchenko, O. Vasylyk, R. Yamnenko, V. Zubchenko
SPECIALLY INVITED PLENARY SPEAKERS Martin Grothaus, Technische Universit¨at Kaiserslautern, Germany
Domenico Marinucci, University of Rome “Tor Vergata”, Italy Mark Podolskij, Aarhus University, Denmark
Mikl´os R´asonyi, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary Rene Schilling, Technical University of Dresden, Germany
SPECIALLY INVITED LECTURE Nicolai Krylov, University of Minnesota, USA
INVITED SPEAKERS FOR PLENARY AND SECTION TALKS Antoine Ayache, University of Lille, France
Ehsan Azmoodeh, University of Liverpool, UK Sandor Baran, University of Debrecen, Hungary
Andreas Basse-O’Connor, Aarhus University, Denmark Luisa Beghin, University of Rome “La Sapienza”, Italy Wolfgang Bock, University of Kaiserslautern, Germany Taras Bodnar, Stockholm University, Sweden
Dariusz Buraczewski, University of Wroclaw, Poland Jose Manuel Corcuera, University of Barcelona, Spain Alexei Daletskii, University of York, United Kingdom
Alessandro De Gregorio, Sapienza University of Rome, Italy Antonio Di Crescenzo, University of Salerno, Italy
Giulia Di Nunno, University of Oslo, Norway Mirko D’Ovidio, Sapienza University of Rome, Italy Istv´an Fazekas, University of Debrecen, Hungary Dmitri Finkelshtein, Swansea University, UK Rita Giuliano-Antonini, University of Pisa, Italy Maria Ivette Gomes, University of Lisbon, Portugal
Alexander Iksanov, Taras Shevchenko National University of Kyiv, Ukraine Peter Imkeller, Humboldt University of Berlin, Germany
Alexander Ivanov, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine Adam Jakubowski, Nicolaus Copernicus University, Poland
Yuri Kondratiev, Bielefeld University, Germany, National Pedagogical Dragomanov University, Ukraine Nino Kordzakhia, Macquarie University, Australia
Volodymyr Koshmanenko, Institute of Mathematics of NAS of Ukraine, Ukraine Kestutis Kubilius, Vilnius University, Lithuania
Alexander Kukush, Taras Shevchenko National University of Kyiv, Ukraine Alexei Kulik, Wroclaw University of Science and Technology, Poland
Nikolay Kuznetsov, V. M. Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine Evgeniy Lebedev, Taras Shevchenko National University of Kyiv, Ukraine
Remigijus Leipus, Vilnius University, Lithuania
Nikolai Leonenko, Cardiff University, UK
Nikolaos Limnios, Compi´egne University of Technology, France
Rostislav Maiboroda, Taras Shevchenko National University of Kyiv, Ukraine Tapabrata Maiti, Michigan State University, USA
Bastien Mallein, Universit´e Sorbonne Paris Nord, France
Igor Malyk, Yuriy Fedkovych Chernivtsi National University, Ukraine Barbara Martinucci, University of Salerno, Italy
Alexander Marynych, Taras Shevchenko National University of Kyiv, Ukraine Ivan Matsak, Taras Shevchenko National University of Kyiv, Ukraine
Matthias Meiners, Mathematisches Institut, Justus Liebig University Giessen, Germany Yuliya Mishura, Taras Shevchenko National University of Kyiv, Ukraine
Ostap Okhrin, TU Dresden, Germany
Andriy Olenko, La Trobe University, Australia
Enzo Orsingher, University of Rome “La Sapienza”, Italy Etienne Pardoux, Aix-Marseille Universit´e, France
Vladimir Piterbarg, Lomonosov Moscow State University, Russian Federation Federico Polito, University of Turin, Italy
Christophe Pouet, ´Ecole centrale de Marseille, France
Mykola Pratsiovytyi, National Pedagogical Dragomanov University, Ukraine Enrico Priola, University of Pavia, Italy
Michael R¨ockner, Bielefeld University, Germany Sylvie Roelly, University of Potsdam, Germany
Barbara R¨udiger, Bergische Universit¨at Wuppertal, Germany
Igor Samoilenko, Taras Shevchenko National University of Kyiv, Ukraine
Mykhaylo Savchuk, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine Jonas ˇSiaulys, Vilnius University, Lithuania
Dmitrii Silvestrov, Stockholm University, Sweden Tommi Sottinen, University of Vaasa, Finland
Evgeny Spodarev, Universit¨at Ulm: Institut f¨ur Stochastik, Germany Peter Spreij, University of Amsterdam, Radboud University, The Netherlands Lukasz Stettner, Institute of Mathematics of the Polish Academy of Sciences, Poland Anatoliy Swishchuk, University of Calgary, Canada
Stefan Tappe, Albert Ludwig University of Freiburg, Germany
Miklos Telek, Budapest University of Technology and Economics, Hungary Grigoriy Torbin, National Pedagogical Dragomanov University, Ukraine Ciprian Tudor, Universit´e de Lille, France
Mikhail Urusov, University of Duisburg-Essen, Germany Alexander Veretennikov, University of Leeds, UK Lauri Viitasaari, University of Helsinki, Finland Andrei Volodin, University of Regina, Canada Nakahiro Yoshida, University of Tokyo, Japan Silvelyn Zwanzig, Uppsala University, Sweden
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CONTENTS
STOCHASTIC DYNAMICS . . . 6
EVOLUTION SYSTEMS, BRANCHING PROCESSES AND RENEWAL THEORY . . . 17
RELIABILITY, QUEUEING AND INFORMATION SECURITY . . . 31
ADVANCED TOPICS IN STOCHASTIC PROCESSES, FRACTIONAL AND RELATED MODELS . . . 42
STOCHASTIC OPTIMIZATION . . . 63
FRACTAL ANALYSIS . . . 70
RISK PROCESSES AND ACTUARIAL MATHEMATICS . . . 78
STATISTICS OF STOCHASTIC PROCESSES AND RANDOM FIELDS . . . 82
HIGH-DIMENSIONAL STATISTICAL INFERENCE . . . 99
STOCHASTIC ANALYSIS AND STOCHASTIC DIFFERENTIAL EQUATIONS . . . 103
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ST OCHAST IC DY N AM ICS
Development of research ideas of Anatoliy Skorokhod Chairs Yu. Kondratiev, W. Bock, V. Koshmanenko, A. Veretennikov
ANATOLII VOLODYMYROVYCH SKOROKHOD (10.09.1930 – 3.01.2011)
Professor Anatolii Volodymyrovych Skorokhod is one of the most outstanding mathematicians of our time. He was a worldwide recognized authority in probability theory, theory of stochastic processes, stochastic analysis, and mathematical statistics. His discoveries – the Skorokhod topology, the Skorokhod integral, Skorokhod’s embedding theorem, the method of common probability space – which entered into practice of the specialists in probability worldwide, have become standard instruments in the probabilist’s toolbox. The theory of random processes achieved its actual level largely due to his contributions. His mathematical work featured beautiful and original ideas, which have since become classical. His scientific and pedagogical activities had a considerable impact on the development of the mathematical culture in the second half of the 20th century.
Anatolii Volodymyrovych Skorokhod was born in September 10, 1930, in Nikopol, an industrial city in the south of Ukraine. His parents were school teachers. His father, Volodymyr Oleksiyovych, taught mathematics, physics and astronomy. His mother, Nadiya Andriivna, taught history, literature and music, besides mathematics. According to Nadiya Andriivna’s memories, their children (two sons) grew up in the aura of the various interests of their parents, who tolerated and respected children, regarding them as equals. The parents paid a lot of attention to their children, stimulating their curiosity, encouraging love to literature, music and poetry, teaching them to live in harmony with nature. This treatment resulted in an early and versatile education of the children, and probably, influenced them both in choosing a scientific career (the younger son, Valerii, made a prominent scientific career in physics, becoming a Full Member of the Ukrainian Academy of Sciences).
In 1935 the Skorokhods settled in Marganets, a town close to Nikopol. Here, in 1937, Anatolii went to primary school. After the Nazi invasion of the Soviet Union in 1941, Anatolii was forced to continue his education at home.
Having survived the horrors of the war, Anatolii’s family moved to Kovel, a town in the Volyn region in the north west of Ukraine, in 1946. The life in Kovel was quite different from the southern regions. Anatolii was impressed by this new environment, where the old national culture and folk traditions were kept alive. Especially, he enjoyed polyphonic choral singing, which was very popular in that region. Anatolii had romantic ideas at that time, he dreamed to become a sea captain. This profession comprised his three passions: mathematics, astronomy and the sea. However, at a medical examination it was discovered that his eyesight was poor and therefore his dream could not be realized, fortunately for mathematics. In 1948, Anatolii completed his secondary education in Kovel, graduating with a gold medal. In the same year he enrolled at the Mechanics and Mathematics Faculty of Kyiv State University.
Skorokhod’s ability to conduct research and his scientific taste first manifested themselves when he was a student.
He started actively working on some problems of mathematical analysis and probability theory. His first research was supervised by B. V. Gnedenko and I. I. Gikhman. Later on, Gikhman also became his colleague and close friend. In his student years, Anatolii solved several problems mostly concerning stable laws. He graduated from the University in 1953 being already an author of five scientific papers.
After graduating from Kyiv University Skorokhod continued his research as a postgraduate student at Moscow State University under the supervision of E. B. Dynkin. It was a period of rapid development of probability theory, and this development was largely due to the probability school at Moscow State University. Fundamental ideas were already contained in A.N. Kolomogorov’s monograph “Grundbegriffe derWahrscheinlichkeitsrechnung” (“Foundations of the theory of probability”) from 1933, studies in the field of limit theorems were conducted, later summarized in the book of B.V. Gnedenko and A. N. Kolmogorov “Limit theorems for sums of independent random variables" (1949), E.
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B. Dynkin investigated Markov processes. All these novel concepts and theories stimulated the creative atmosphere that prevailed at scientific seminars led by A. N. Kolmogorov and E. B. Dynkin, attracting and bringing together talented young people from different regions of the former Soviet Union. And among them, the young scientist from Kyiv was distinguished by his profound knowledge and a lot of original approaches and unexpected ideas.
Skorokhod’s work of this period already contained some revolutionary concepts and methods in the theory of stochastic processes. He introduced a new topology on the space of c´adl´ag functions (right-continuous with left limits), which he called J -topology. However, in the scientific literature it is usually referred to as Skorokhod topology, and the corresponding topological space is called Skorokhod space. Skorokhod commented on the advantages of the J -topology as follows: “. . . the uniform topology in D[0, 1] means that if xn(t) converges to x(t), then for n large enough the points of discontinuity of xn(t) coincide with those of x(t). This means that if t is regarded as time parameter, we must assume the existence of a tool capable to measure time exactly, which is physically impossible. It is much more natural to suppose that the functions which are obtained from each other by a small deformation of the time scale lie close to each other. We thus are led to propose the topology J ”.
In 1956, a new scientific journal, “Theory of probability and its applications”, was founded. Its first volume contained Skorokhod’s paper “Limit theorems for stochastic processes”, which gave a substantial generalization of Donsker’s invariance principle to a much wider class of random objects. This publication became well-known to the scientific community, and since this time Skorokhod’s work has played an important role in the development of the theory of random processes; to a great extent, his ideas laid the foundations for further investigations in this theory worldwide.
In the same paper, another fundamental idea was proposed, now known as the method of common probability space. The crucial point of this method is that a weakly converging sequence of random elements can be replaced by a strongly converging sequence of elements with the same distribution defined on a common probability space.
In 1956, Skorokhod defended his Candidate Thesis (equivalent to a Ph.D.) “Limit theorems for random processes”
and returned to Kyiv, where he began lecturing at Kyiv University. At the same time he conducted intensive scientific research. Skorokhod studied the limit behavior of functionals of stochastic processes and obtained estimates for the rate of convergence in limit theorems. To estimate the probabilities that a sequence of normalized sums of independent identically distributed random variables lies between certain boundaries, Skorokhod suggested an original method. The idea was to replace the sequences of the sums by the values of Brownian motion at certain random instants of time such that the distributions of both sequences coincide. Later on, Skorokhod’s original idea was generalized and this fundamental method, now referred to as “Skorokhod embedding”, became a standard tool for studying sums of random variables.
At the same time Skorokhod started research on the existence and uniqueness of solutions to stochastic differential equations, including those with boundary conditions. He also investigated the differentiability of measures correspond- ing to stochastic processes, and developed statistical methods for stochastic processes. In 1961, his first monograph,
“Studies in the theory of random processes (Stochastic differential equations and limit theorems for Markov processes)”
was published in Russian, and it was translated to English in 1965. In this book, Skorokhod widely used his purely probabilistic methods of proof. In the preface, Skorokhod 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 univer- sality, while the advantage of probabilistic methods is their intrinsic relation to the problem. This book was the basis for his thesis “Stochastic differential equations and limit theorems for random processes”, which, in 1962, earned him the degree of Doctor of Sciences (an analogue of the habilitation). In his next monograph “Stochastic processes with independent increments” (1964), Skorokhod gave a systematic description of homogeneous stochastic processes with independent increments, established limit theorems for them and gave criteria for absolute continuity and singularity of measures. The second edition of this book was published in 1986 and translated to English in 1991. In 1964, Sko- rokhod became chair of the new Department of the Theory of Random Processes at the Institute of Mathematics of the Ukrainian Academy of Sciences, while continuing giving lectures at Kyiv University. In 1965, Skorokhod published the new monograph “Introduction to the theory of random processes” (jointly written with I. I. Gikhman). This book was translated by several publishers and quickly became a classic on the theory of stochastic processes. During this period, Skorokhod investigated the local structure of continuous Markov processes, proved non-classical limit theo- rems for random walks, studied multiplicative families of random operators, multiplicative functionals and nonlinear transformations of random processes. An important part of his research was concerned with stochastic differential equations. Using his purely probabilistic methods, Skorokhod obtained several significant results. The research in this area culminated in the monograph “Stochastic differential equations”, jointly written with I. I. Gikhman, published in 1968 and translated to German in 1971 and to English in 1972. In 1970 this monograph gained the Award of the Ukrainian Academy of Sciences.
Another important direction of his research in the theory of stochastic differential equations was related to equations that describe processes on manifolds with boundary. Here Skorokhod obtained pioneering results, which stimulated many mathematicians all over the world to investigate similar problems. Due to his great impact in this area, equations which describe diffusions in a domain with reflecting boundary are usually referred to as Skorokhod equations. For his scientific achievements, Skorokhod was elected Corresponding Member of the Ukrainian Academy of Sciences in 1967.
Skorokhod always distinguished himself by his independent opinion. He always kept his beliefs and moral standards, though this was quite dangerous under the totalitarian regime. In 1968, he joined the campaign of a group of Ukrainian intellectuals defending the constitutional rights of the citizens of the country. All participants of this group were punished. Skorokhod was forbidden to give lectures at the university, to advise students, and he was excluded from the editorial boards of some scientific journals. For fifteen years his attempts to participate in scientific conferences abroad failed, as he was not allowed to leave the Soviet Union. To his credits, Skorokhod bore his misfortunes bravely and during this period his scientific output was exceptional: 10 monographs, including the three-volume gargantuan (around 1800 pages) “Theory of stochastic processes” (1971–1975, jointly written with I. I. Gikhman), over 100 scientific articles, a dozen of textbooks and popular scientific books.
Such a high scientific productivity combined with Skorokhod’s absence at international scientific conferences was some kind of a mystery. Some foreign scientists believed that “Skorokhod” was a collective name of Soviet experts working in the field of the theory of random processes, just as a group of French scientists united under the name
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“Bourbaki”. In 1980 the famous French mathematician Paul Malliavin came to Kyiv. He said that he was glad to know that such a person really existed.
The three-volume monograph “The theory of stochastic processes” (1971–1975) gave a thorough systematic overview of the theory of stochastic processes. Twenty years after Doob’s “Stochastic processes” was published, this monograph pursued a similar objective. Since that time the theory of stochastic processes experienced several revolutionary changes (some of them due to Gikhman and Skorokhod), and the scientific community appreciated the monograph as a very timely work. Immediately after its appearance the monograph was translated and became an extremely popular concise course on the theory of stochastic processes, aiming at an advanced reader. Later on, in 1982, Skorokhod was awarded the State Prize of Ukraine in Science and Technology for this monograph.
At the same time, Skorokhod turned to various problems of measure theory; he investigated questions concerning the convergence, absolute continuity and transformation of measures, and integration in abstract spaces. The results of these investigations are summarized in the monograph “Integration in Hilbert space” (1974). Skorokhod also considered problems of martingale theory and, more importantly, questions of stochastic integration. In 1975, Skorokhod made another major breakthrough – he found a way to integrate non-adapted processes with respect to a Wiener process, thus generalizing the Ito integral. Later his approach was vastly generalized and applied in a variety of situations to different processes, even to those neither having the martingale nor the Markov properties. In view of Skorokhod’s great pioneering contribution to this topic, this construction is usually called the Skorokhod integral, despite the fact that such a generalization was found independently by Hitsuda.
Parallel to these research activities he published the monographs “Controlled stochastic processes” (1977, jointly written with I. I. Gikhman), “Random linear operators” (1978), “Consistent estimates of parameters of random pro- cesses” (1980, jointly written with I. Sh. Ibramkhalilov). The theories described in these monographs were developed by him from scratch and brought to a level of depth, which were already suitable for numerous applications. Since the beginning of the 1980s, Skorokhod intensified the communication with foreign scientists. His period of forced isolation ended, and he started to deliver talks at major international events, which attracted a lot of interest by the scientific community. Most notable are his participations as plenary speaker at the First International Bernoulli Society Congress (1986), the International Mathematical Congress (1986), the Paul L´evy Colloquium on Stochastic Processes (1987), where he also was a co-organizer, and at the Second International Bernoulli Congress (1990), where he chaired a section.
His scientific interests of that period were closely related to the elaboration of the ideas which appeared in his earlier work. Most attention was given to stochastic differential equations. His findings were included in the monograph
“Stochastic differential equations and their applications” (1982, jointly written with I. I. Gikhman). The questions considered therein include integration with respect to random measures, weak solvability of stochastic differential equations, limits theorems, Markov processes. In the following monograph “Stochastic equations for complex systems”
(1983) he studied more complex questions, such as stochastic differential equations on manifolds with a boundary in spaces with branching points, where the dimensions may change, and the limit behavior of a large system of randomly interacting particles.
In 1985, Skorokhod became a Full Member of the Ukrainian Academy of Sciences. In 1982 and in 2003, he was awarded the Ukrainian State Prize in Science and Technology.
Since 1993, Skorokhod worked at Michigan State University (Lansing, Michigan, USA), retaining close scientific relations with the Institute of Mathematics of the Ukrainian Academy of Sciences. His scientific work of these last years was devoted for the investigation of the asymptotic behavior of dynamical systems under random perturbations. The 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). At that time, he also wrote, together with Shlomo Leventhal, a highly reputed article on financial mathematics, “On the possibility of hedging options in the presence of transaction costs”. In 2000, he became a Member of the American Academy of Art and Science.
As a person, Anatolii Volodymyrovych Skorokhod was always punctual, well organized. His ability to concentrate his thoughts was exceptional. He wrote reviews and other documents in one sitting, without corrections. Skorokhod was never in a hurry, he never refused to advise students, when they asked. He never complained about the lack of time. According to his relatives, he did his main research work from 6 am to 8 am, never allowing himself to give up these morning hours. Skorokhod has always been extremely modest, but the feeling of inner freedom had never left him. And he taught this inner freedom to his students and colleagues, repeatedly stressing that it is important not only to be a good expert, but first of all a decent person.
From the very beginning of his work at Kyiv University, Skorokhod distinguished himself by a unique manner of delivering lectures. Skorokhod’s lectures at Kyiv University were both deep and accessible for students. He lived in a constant state of creativity. It was not uncommon for him to prove some statements impromptu at the lectures. In this way, he made his students participants of a creative scientific process. During special courses Skorokhod sometimes delivered his own fresh results, even unpublished yet. When Skorokhod returned to Kyiv from Moscow in 1956, the life of the scientific seminar on probability theory at Kyiv University became much more active. His discussions with speakers and his capability to understand the core of a problem, to generalize it, to find possible weak points in the proof, and to reveal the hidden relation to other problems turned the seminar sessions into a really creative laboratory.
In view of this, all interested scholars tried to deliver a talk at the seminar in the presence of Skorokhod. This was very instructive not only for the speaker but also for the participants at the seminar. Skorokhod often talked about his new research, offering many new problems to solve. Thus, the Kyiv probability school was to a great extent the result of Skorokhod’s activities, and it is hard to overestimate his role in the development of this school. He had over 50 disciples, 17 of which became doctors of sciences (habilitated doctors).
A.V. Skorokhod was the Founding Editor and Editor-in-Chief in 1970–2011 of the journal Theory of Probability and Mathematical Statistics. Being established firstly as a leading journal on probability and statistics in Ukraine, the journal has been developed by now into respectable international journal indexed in Scopus and Web of Science. After its 50 years long history, starting with Issue 102, 2020, the journal is published jointly by Taras Shevchenko National University of Kyiv and American Mathematical Society as an original journal in English.
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Skorokhod always paid considerable attention to the mathematical education of the youth and to spreading math- ematical knowledge as widely as possible. Skorokhod wrote, in collaboration with his colleagues, several elementary textbooks and popular science books. To name a few, these are “Selected topics of elementary mathematics”, which had four editions, “Elements of probability theory and random processes” (three editions), and “Theory of probability.
Collection of problems” (three editions including an English translation). He popularized the names of distinguished Ukrainian mathematicians of the past: Georgiy Voronoi, Victor Bunyakovskii, Mykhailo Kravchuk, and others. With this goal in mind, he undertook several journeys to lecture all over Ukraine.
Skorokhod was the author of 24 scientific monographs, most of which were translated, and more than 300 articles were published in scientific journals.
Selected monographs and textbooks
[1] A. V. Skorokhod. Studies in the theory of random processes. Stochastic differential equations and Markov processes. (Issledovaniya po teorii sluchainykh protsessov. Stokhasticheskie differentsialnye uravneniya i pre- delnye teoremy dlya protsessov Markova.). Izdat. Kiev. Univ., Kiev, 1961.
[2] A. V. Skorokhod. Stochastic processes with independent increments. (Sluchainye protsessy s nezavisimymi prirashcheniyami.). “Nauka”, Moscow, 1964.
[3] I. I. Gikhman and A.V. Skorokhod. Vvedenie v teoriyu sluchainykh protsessov. (Introduction to the theory of random processes). “Nauka”, Moscow, 1965.
[4] A. V. Skorokhod. Studies in the theory of random processes. Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. Translated from the Russian by Scripta Technica, Inc.
[5] I. I. Gikhman and A. V. Skorokhod. Stochastic differential equations. (Stokhasticheskie differentsialnye uravneniya.). “Naukova Dumka”, Kiev, 1968.
[6] A. Skorokhod and N. Slobodenyuk. Limit theorems for random walks. (Predel’nye teoremy dlja slucainykh bluzdanii.). “Naukova Dumka”, Kiev, 1970.
[7] I. I. Gikhman and A. V. Skorokhod. The theory of stochastic processes. Vol. I. (Teoriya sluchainykh protsessov.
Tom I.). “Nauka”, Moscow, 1971.
[8] I. I. Gikhman and A. V. Skorokhod. Stochastic differential equations, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 72. Springer-Verlag, New York, 1972. Translated from the Russian by Kenneth Wickwire.
[9] I. I. Gikhman and A. V. Skorokhod. The theory of stochastic processes. Vol. II. (Teoriya sluchainykh protsessov. Tom II.). “Nauka”, Moscow, 1973.
[10] I. I. Gikhman and A. V. Skorokhod. The theory of stochastic processes. I, Die Grundlehren der mathema- tischen Wissenschaften, vol. 210. Springer-Verlag, Berlin-Heidelberg-New York, 1974. Translated from the Russian by S. Kotz.
[11] A. V. Skorokhod. Integration in Hilbert space, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 79.
Springer-Verlag, New York, 1974.
[12] I. Gikhman and A. Skorokhod. Theory of random processes. Vol. III. (Teoriya sluchajnykh processov. Tom III.). “Nauka”, Moscow, 1975.
[13] I. I. Gikhman and A.V. Skorokhod. The theory of stochastic processes. II, Die Grundlehren der mathematis- chen Wissenschaften, vol. 218. Springer-Verlag, Berlin-Heidelberg-New York, 1975.
[14] I. Gikhman and A. Skorokhod. Controlled stochastic process. (Upravljaemye sluchainye processy.). “Naukova Dumka”, 1977.
[15] I. Gikhman and A. Skorokhod. Introduction to the theory of random processes. 2nd ed., revised. (Vvedenie v teoriyu sluchajnykh protsessov.). “Nauka”, Moscow, 1977.
[16] A. Skorokhod. Limit theorems for random processes. (Predel’nye teoremy dlia sluchainykh processov.). Izd.
Inst. Mat. Acad. Nauk Ukr. SSR., Kiev, 1977.
[17] V. S. Korolyuk, N.I. Portenko, A.V. Skorokhod, A.F. Turbin. A manual on probability theory and mathe- matical statistics. (Spravochnik po teorii veroyatnostei i matematicheskoi statistike.). “Naukova Dumka”, Kiev, 1978.
[18] A. V. Skorohod. Random linear operators. (Sluchainye lineinye operatory.). “Naukova Dumka”, Kiev, 1978.
[19] I. Gikhman, A. Skorokhod. Controlled stochastic processes. Springer-Verlag, Berlin-Heidelberg-NY, 1979.
[20] I. I. Gikhman and A.V. Skorokhod. The theory of stochastic processes. III. Springer-Verlag, Berlin-Heidelberg- New York, 1979, iii+387 pp.
[21] A. J. Dorogovcev, D. S. Sil’vestrov, A. V. Skorokhod, and M. I. Yadrenko. Probability theory: a collection of problems. (Teoriya veroyatnostei. Sbornik zadach.). “Vyscha Shkola”, Kiev, 1980.
[22] I. I. Gikhman and A. V. Skorokhod. Stochastic differential equations and their applications. (Stokhasticheskie differentsialnye uravneniya i ikh prilozheniya.). “Naukova Dumka”, Kiev, 1982.
[23] A. V. Skorokhod. Stochastic equations for complex systems. (Stokhasticheskie uravneniya dlya slozhnykh sistem.). ”Nauka”, Moscow, 1983.
[24] A. V. Skorokhod. Random linear operators. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, 1984, xvi+199 pp.
[25] V. S. Korolyuk, N. I. Portenko, A. V. Skorokhod, and A. F. Turbin. A manual on probability theory and mathematical statistics. 2nd ed. (Spravochnik po teorii veroyatnostej i matematicheskoj statistike.). “Nauka”, Moscow, 1985.
[26] A. V. Skorokhod. Stochastic processes with independent increments. 2nd ed. (Sluchainye protsessy s nezav- isimymi prirashcheniyami.). “Nauka”, Moscow, 1986.
[27] A. V. Skorokhod. Asymptotic methods in the theory of stochastic differential equations. (Asimptoticheskie metody teorii stokhasticheskikh differentsial’nykh uravnenij.). “Naukova Dumka”, Kiev, 1987.
[28] A. V. Skorokhod. Stochastic equations for complex systems, Mathematics and its Applications (Soviet Series), vol. 13. D. Reidel Publishing Co., Dordrecht, 1988.
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[29] A. V. Skorokhod. Asymptotic methods in the theory of stochastic differential equations, Translations of Mathematical Monographs, vol. 78. American Mathematical Society, Providence, RI, 1989.
[30] A. V. Skorohod. Random processes with independent increments, Mathematics and its Applications (Soviet Series), vol. 47. Kluwer Academic Publishers Group, Dordrecht, 1991.
[31] A. V. Skorokhod. Lectures on the theory of stochastic processes. VSP, Utrecht, 1996.
[32] A. Y. Dorogovtsev, D. S. Silvestrov, A. V. Skorokhod, and M. I. Yadrenko. Probability theory: collection of problems, Translations of Mathematical Monographs, V. 163. American Mathematical Society, Providence, RI, 1997.
[33] A. V. Skorokhod, F. C. Hoppensteadt, and H.Salehi. Random perturbation methods with applications in science and engineering, Applied Mathematical Sciences, vol. 150. Springer-Verlag, New York, 2002.
[34] A. V. Skorokhod. Basic principles and applications of probability theory. Springer-Verlag, Berlin, 2005.
STOCHASTIC CAMASSA-HOLM EQUATION WITH CONVECTION TYPE NOISE
Sergio Albeverio1, Zdzis law Brze´zniak2, Alexei Daletskii3 The (deterministic) Camassa-Holm equation is a non-local partial differential equation describing propagation of waves in shallow water. Although fist introduced by B. Fuchssteiner and A. Fokas in 1981 as a member of a family of integrable Hamiltonian equations, it was rediscovered by R. Camassa and D. Holm (Phys. Rev. Letters, v. 71, 1993), who gave its physical derivation and interpretation. In contrast to the Korteveg-de-Vries equation, the Camassa-Holm equation admits so-called peaked solutions describing wave breaking phenomena. Various aspects of the Camassa-Holm equation have been extensively studied.
We consider a stochastic version of the Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as the diffusion coefficient, which was introduced by D. Crisan and D. Holm (Physica D: Nonlinear Phenomena, v. 376-377, 2018). We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato’s operator theory methods. Some of the results have potential to find applications to other nonlinear stochastic partial differential equations.
It is our honor to dedicate this work to the memory of Anatoliy Skorokhod, as a small gratitude for his influence and pioneering research in stochastic analysis.
References
[1] Albeverio, S., Brze´zniak, Z., Daletskii, A., Stochastic Camassa-Holm equation with convection type noise, arXiv:1911.07077 [math.FA].
1 Institute for Applied Mathematics, Rheinische Friedrich-Wilhelms Universit ¨at Bonn and Hausdorff Center for Mathematics, Bonn, Germany
E-mail address: [email protected]
2 Department of Mathematics, University of York, United Kingdom E-mail address: [email protected]
3 Department of Mathematics, University of York, United Kingdom E-mail address: [email protected]
STRONG KAC’S CHAOS IN THE MEAN-FIELD BOSE-EINSTEIN CONDENSATION
S. Albeverio1, A. Romano2, F. C. De Vecchi3, S. Ugolini4 A stochastic approach to the (generic) mean-field limit in Bose-Einstein Condensation is briefly described as well as the convergence of the ground state energy in the termodynamic limit. A strong form of Kac’s chaos on path-space for the k-particles probability measures are derived from the previous energy convergence by purely probabilistic techniques notably using a simple chain-rule of the relative entropy. The Fisher’s information chaos of the fixed-time marginal probability density under the generic mean-field scaling limit and the related entropy chaos result are also deduced.
References
[1] Albeverio, S., De Vecchi, F.C., Romano, A., Ugolini, S., Strong Kac’s chaos in the mean-field Bose-Einstein Condensation. Stochastics and Dynamics 20 (5), 2020, 2050031.
1Institute for Applied Mathematics and Hausdorff Center for Mathematics, Rheinische Friedrich-Wilhelms-Universit ¨at Bonn, Endenicher Allee 60, 53115 Bonn, Germany
2 Dipartimento di Matematica, Universit `a di Milano, Via Saldini 50, Milano, Italy
3Institute for Applied Mathematics and Hausdorff Center for Mathematics, Rheinische Friedrich-Wilhelms-Universit ¨at Bonn, Endenicher Allee 60, 53115 Bonn, Germany
4 Dipartimento di Matematica, Universit `a di Milano, Via Saldini 50, Milano, Italy E-mail address: [email protected]
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DELAY-INDUCED PERIODIC BEHAVIOUR IN COMPETITIVE POPULATIONS
M. Aleandri1, I. G. Minelli2 We study a model of binary decisions in a fully connected network of interacting agents. Individual decisions are determined by social influence, coming from direct interactions with neighbours, and a group level pressure that accounts for social environment. We study the convergence of the process as the number of agents goes to infinity and the propagation of chaos. Moreover, when the number of agents are large but fixed, we show the amount of time spent by the process around the stable points of the macroscopic dynamics. As a consequence, in a competitive environment, the interplay of these two aspects results in the presence of a persistent disordered phase where no majority is formed.
We show how the introduction of a delay mechanism in the agent’s detection of the global average choice may drastically change this scenario, giving rise to a coordinated self sustained periodic behaviour. When the delay mechanism is led by kernels of Erlang type the limit dynamics is described by a finite number of equations. In the competitive environment tuning the parameters of the delay an Hops bifurcation occurs and limit cycles appear. Through the paper we show the evolution of the microscopic process performing some simulations.
1 Luiss University, Viale Romania, 32, 00197 Rome, Italy E-mail address: [email protected]
2Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universit `a degli Studi dell’Aquila, Via Vetoio (Coppito 1), 67100 L’Aquila, Italy
E-mail address: [email protected]
GENERALIZED SCALING OPERATORS AND GENERALIZED GREY BROWNIAN MOTION – A GENERALIZATION BEYOND GAUSSIANITY
W. Bock In this talk we will introduce a generalization of Brownian motion which leads to a class of in general non- Gaussian processes. We present results obtained for this process and give a brief overview about its properties. In addition we give a representation using grey Ornstein-Uhlenbeck processes. An ergodicity breaking parameter shows why the process is challenging from the view of numerical simulations. The Mittag-Leffler Analysis was introduced by Grothaus et. al., generalizing properties of the Gaussian measure to the non-Gaussian setting with the help of generalized Appell systems. We will study generalized scaling operators in this setting and relate them to the concept of generalized Mittag-Leffler kernels. An outlook to transformation groups is given.
Department of Mathematics, TU Kaiserslautern, Germany E-mail address: [email protected]
STOCHASTIC DIFFERENTIAL EQUATIONS IN A SCALE OF HILBERT SPACES AND INFINITE PARTICLE DYNAMICS
Georgy Chargaziya1, Alexei Daletskii2 We consider a stochastic differential equation in a scale of densely embedded Hilbert spaces {Xα}α∈[a,b], Xα⊂ Xβ
for a ≤ α < β ≤ b, and assume that the coefficients satisfy the following generalized Lipschitz-type condition: there exists 0 < q <12 such that
||f (x) − f (y)||β≤ L
(β − α)q||x − y||α f or all α < β ∈ [a, b].
Our main example is an infinite system of one- (or finite-) dimentional stochastic equations with coefficients depending on finite but unbounded number of components. Such systems of equations are realated to the study of particle systems on random (e.g. Poisson-distributed) configurations in Euclidean spaces, see [1]. We show that, for an initial data in Xα, there exists a unique strong solution with infinite lifetime in any Xβ with β > α. We build upon the methods of papers [2–4], our main technical tool being a modified version of Ovsyannikov’s method.
References
[1] Daletskii, A., Kondratiev, Yu., Kozitsky, Yu., Pasurek, T., Gibbs states on random configurations. Journal of Mathematical Physics 55, 2014, 083513.
[2] Daletskii, A., Stochastic differential equations in the scale of Hilbert spaces. Electronic Journal of Probability 23 (1), 2018, 1–15.
[3] Daletskii, A., Finkelshtein, D., Non-equilibrium particle dynamics with unbounded number of interacting neighbors. Journal of Statis- tical Physics 122 (1), 2018, 1–23.
[4] Albeverio, S., Kondratiev, Yu., Rockner, M., Tsikalenko, T., Glauber dynamics for quantum lattice systems. Reviews in Mathematical Physics 13 (1), 2001, 51–124.
Mathematics Department, University of York, United Kingdom E-mail address: 1[email protected], 2[email protected]
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PATH DEPENDENT KYLE EQUILIBRIUM MODEL
Jos´e Manuel Corcuera We study the equilibrium in the model proposed by Kyle in 1985 and extended to the continuous time setting by Back in 1992. The novelty of this paper is that we consider a general price functional of the path of the aggregate demand and by using the functional Itˆo calculus we give necessary and sufficient conditions for the existence of an equilibrium. We also study the equilibrium when the insider is risk averse.
References
[1] Back, K., Insider trading in continuous time. The Review of Financial Studies 5 (3), 1992, 387–409.
[2] Kyle, A. S., Continuous auctions and insider trading. Econometrica 53 (6), 1985, 1315–1335.
Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, E-08007 Barcelona, Spain E-mail address: [email protected]
The work of J. M. Corcuera is supported by the Spanish grant MTM2016-76420-P.
SENSITIVITY ANALYSIS IN THE INFINITE DIMENSIONAL HESTON MODEL
G. Di Nunno We consider the infinite dimensional Heston stochastic volatility model. The price of a forward contract on a non- storable commodity is modelled by a generalised Ornstein-Uhlenbeck process in the Filipovic space with this volatility.
We prove a representation formula for the forward price. Then we consider prices of options written on these forward contracts and we study sensitivity analysis with computation of the Greeks with respect to different parameters in the model. Since these parameters are infinite dimensional, we need to reinterpret the meaning of the Greeks. For this we use infinite dimensional Malliavin/Skorokhod calculus and a randomisation technique. The presentation is based on joint work with Fred Espen Benth and Iben Simonsen.
References
[1] Benth, F. E., Di Nunno, G., Simonsen, I. C., Sensitivity analysis in the infinite dimensional Heston model. arXiv:2012.12167.
Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway E-mail address: [email protected]
This research is supported by the Research Council of Norway within the project nr 274410, STORM - Stochastics for Time-Space Risk Models.
SPATIO-TEMPORAL CORRELATIONS IN INDIVIDUAL-BASED MODELS
Dmitri Finkelshtein Individual-based models are used to study complex phenomena in many fields of science. While simulating agent- based models is often straightforward, predicting their behaviour mathematically has remained a key challenge. Re- cently developed mathematical methods allow the prediction of the emerging spatial patterns for a general class of agent-based models, whereas the prediction of spatio-temporal pattern has been thus far achieved only for special cases.
We present a general technique that allows deriving the spatio-temporal (pair) correlation structure for a general class of individual-based models. To do so, we define an auxiliary model, in which each agent type of the primary model expands to three types, called the original, the past and the new agents. In this way, the auxiliary model keeps track of both the initial and current state of the primary model, and hence the spatio-temporal correlations of the primary model can be derived from the spatial correlations of the auxiliary model. We illustrate also the agreement between analytical predictions and agent-based simulations using two example models from theoretical ecology.
References
[1] Ovaskainen, O., Somervuo, P., Finkelshtein, D., A general mathematical method for predicting spatio-temporal correlations emerging from agent-based models. J. Royal Society Interface 17, 2020, 20200655, http://doi.org/10.1098/rsif.2020.0655.
Swansea University, United Kingdom E-mail address: [email protected]
ERGODICITY AND REGULARITY FOR AFFINE PROCESSES
Martin Friesen An affine process is characterized by its log-characteristic function which depends in an affine manner on the initial state of the process, i.e.
Exehu,Xti = exp (φ(t, u) + hx, ψ(t, u)i) , u ∈ iRd, x ∈ D, 12
where φ(t, u) ∈ C, ψ(t, u) ∈ Cd, and D ⊂ Rd denotes the state space of the process. On the canonical state space D = Rm+× Rn, d = m + n > 0, this notion includes Ornstein-Uhlenbeck as well as continuous-state branching processes with immigration and, moreover, unifies them in a general framework.
The study of invariant measures and ergodicity for affine processes has recently received much attention in the literature. In this talk, we first address the existence and uniqueness of invariant measures and then, based on the coupling technique, prove convergence of transition probabilities with exponential rate towards the unique invariant measure in the Wasserstein distance. Afterward, we discuss the regularity (in anisotropic Besov spaces) of correspond- ing heat kernels and deduce from that the strong Feller property. By a combination of this regularity results and the coupling technique, we finally also prove convergence in total variation distance for affine processes.
This talk is based on several joint works with: P. Jin, J. Kremer, and B. R¨udiger.
University of Wuppertal, Germany
E-mail address: [email protected]
RECENT ADVANCES ON THE A. S. SKOROKHOD REPRESENTATION
Adam Jakubowski It took more than twenty years to find first non-trivial applications of the a.s. Skorokhod representation for weakly convergent random elements with values in a Polish space [5] (see e.g. [6] for statistics or [4] for stochastic analysis).
It is interesting that a similar time distance was needed to accept an extension of the Skorokhod construction to non-metric spaces, obtained by the author in 1998, as a convenient and useful tool in existence problems of stochastic partial differential equations.
During the lecture we shall discuss the most important steps in development of Skorokhod’s idea, including so called spaces with Skorokhod representation property [1], [2] and recent constructions in submetric spaces.
References
[1] Banakh, T. O., Bogachev, V. I., Kolesnikov, A. V., Topological spaces with Skorokhod representation property. Ukrainian Math. J. 57, 2005, 1371–1386.
[2] Bogachev, V. I, Kolesnikov, A. V., Open mappings of probability measures and the Skorokhod representation theorem. Theor. Probab.
Appl. 46 (1), 2001, 20–38.
[3] Jakubowski, A., The a.s. Skorohod representation for subsequences in nonmetric spaces. Theory Probab. Appl. 42, 1998, 167–174.
[4] Jakubowski, A., M´emin, J., Pag`es, G., Convergence en loi des suites d’int´egrales stochastiques sur l’espace ID de Skorokhod. Probab.
Theory Related Fields Vol. 81, 1989, 111–137.
[5] Skorohod, A. V., Limit theorems for stochastic processes, Theor. Probab. Appl. 1, 1956, 261–290.
[6] van der Vaart, A., Wellner, J. A. Weak Convergence and Empirical Processes. With Applications to Statistics. Springer Series in Statistics, Springer, 1996.
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toru ´n, Poland
E-mail address: [email protected]
A MODEL OF CONFLICT NETWORK WITH ADDITIVE EXTERNAL INFLUENCES
T. V. Karataieva1, V. D. Koshmanenko2 We study the mathematical models of networks as dynamical conflict systems with repulsive interaction and external influences. In terms of discrete probability measures p = (p1, ..., pi, ..., pm), m > 2 the time evolution is governed by the following equations
pt+1i = (pti+ bti)θt+ 1 − rit
zt , rit= 1 − pti
m − 1, 0 ≤ bti≤ 1, i = 1, m, where θtis a bounded positive function, and zt=P
k(ptk+ btk)(θt+ 1 − rtk) is a normalizing denominator. We show the existence of limits p∞i = limt→∞pti in the case bti= bi, i = 1, m. In particular, if bi= β is the same for all i and
β ≥ 2 maxk=1,m{pk} −Pm k=1pk2
m(m − 2) ,
then p∞= (1/m, ..., 1/m). We observe the existence of limit cyclic states in the case bti = αtipi−1t , ∀i = 2, m, bt1= αt1ptm with appropriate chosen αti.
Our results have the social network interpretation. The external influences bti may be used for management of the individual power and energy in terms of pti and to provide some standard minimal level for all of them.
References
[1] Karataieva, T., Koshmanenko, V., Krawczyk, M., Kulakowski, K., Mean field model of a game for power. Physica A: Statistical Mechanics and its Applications 525, 2019, 535–547.
[2] Karataieva, T., Koshmanenko, V., Social unite as a mathematical model of the conflict dynamical system (in Ukrainian). Nonlinear Oscillations 22 (1), 2019, 66–85.
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska Str. 3, Kyiv 01024, Ukraine E-mail address: 1[email protected], 2[email protected]
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LIFE IN RANDOM TIMES
Yuri Kondratiev The concept of random times does appear in several real world models in the contrast to the Newton time motion usual in classical mechanics. Our aim is to show how a random time will change the behavior of considered systems.
We consider two classes of dynamics. At first, random time Markov processes will be analyzed. Secondly, we study random time deterministic dynamical systems which are (in certain sense) special cases of Markov evolution.
Bielefeld University, Germany E-mail address: [email protected]
ON A FAIR DIVISION OF THE RESOURCE SPACE BETWEEN MANY ALTERNATIVE OPPONENTS
Volodymyr Koshmanenko We study the behavior of complex dynamical systems with conflict interaction between opponents presented by probability measures [1, 2]. More precisely, we discuss the problem of a fair redistribution of a compact Ω (resource space) between many, m ≥ 2, alternative opponents associated with a family of probability measures µi, i ∈ 1, m which subjected to the nonlinear transformation:
d
dtµti= Hµti− ηti,
where H corresponds to the free evolution and ηit stands for the occupation (mean field) measure produced by all µk, k 6= i.
We prove the existence of asymptotic limits
t→∞lim µti = µ∞i , µ∞i ⊥ µ∞k , i 6= k,
which establish the equilibrium division of Ω. This division generalizes the well-known Hahn-Jordan decomposition theorem for signed measures:
Ω = Ω+1 [
· · ·[
Ω+m, supp(µ∞i ) = Ω+i , where Ω+i is the region of total domination of µi over all other µk, k 6= i.
References
[1] Koshmanenko, V., Theorem of conflicts for a pair of probability measures. Math. Methods of Operations Research 59 (2), 2004, 303–313.
[2] Karataieva, T., Koshmanenko, V., Krawczyk, M., Kulakowski, K., Mean field model of a game for power. Physica A 525, 2019, 535–547, https://doi.org/10.1016/j.physa.2019.03.110.
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska Str. 3, Kyiv 01024, Ukraine E-mail address: [email protected]
ON TIME INHOMOGENEOUS STOCHASTIC IT ˆ O EQUATIONS WITH DRIFT IN L
d+1Nicolai Krylov We prove the solvability of Itˆo stochastic equations with uniformly nondegenerate, bounded, measurable diffusion and drift in Ld+1(Rd+1). Actually, the powers of summability of the drift in x and t could be different. Our results seem to be new even if the diffusionВ is constant. The method of proving the solvability belongs to A. V. Skorokhod.
Weak uniqueness of solutions is an open problem even if the diffusion is constant.
University of Minnesota, USA E-mail address: [email protected]
NON MARKOVIAN EPIDEMIC MODELS, LAW OF LARGE NUMBERS AND CENTRAL LIMIT THEOREMS
Etienne Pardoux Most epidemic models are deterministic ODE models, which can be thought of as law of large numbers limits of Markovian individual based stochastic models. The Markov assumption required that the time during which each individual stays in a specific compartment (e.g. remains infectious) follows an exponential distribution, which is quite unrealistic. Also, the ODE type of model lacks memory, when sudden changes happen in the behavior of the society, like the lockdown that many countries in the world have experienced during the last year, in order to face the Covid-19 epidemic. In reality, there is a delay in the effect of the lockdown measures, which appropriate models with memory,
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which are law of large numbers limits of non Markov stochastic models, reproduce properly. We shall describe a class of non Markov models, discuss their LLN limits (which are Volterra integral equations) and the FCLT results. We shall also describe models with varying infectivity, and where immunity is lost gradually. Our results make extensive use of weak convergence techniques in the Skorokhod space. Joint work with Guodong Pang (Penn. State Univ., USA), Rapha¨el Forien (INRAE Avignon, France) and in part Arsene Brice Zotsa-Ngoufack (Phd Student Marseille - Yaounde).
Aix-Marseille Universit´e, France E-mail address: [email protected]
EQUILIBRIA OF NONLINEAR DISTORTED BROWNIAN MOTIONS
Michael R¨ockner Joint work with: Viorel Barbu (Romanian Academy, Iasi)
This talk will review the connection of nonlinear Fokker–Planck–Kolmogorov (FPK) equations and McKean–Vlasov SDEs, with special emphasis on the case where the coefficients depend Nemytskii-type on the time marginal laws. A class of examples are nonlinear distorted Brownian motions. Recent results on their asymptotic behaviour, obtained through their corresponding nonlinear FPK equations, will be presented.
References
[1] Barbu, V., R¨ockner, M., Probabilistic representation for solutions to nonlinear Fokker-Planck equations. SIAM J. Math. Anal. 50 (4), 2018, 4246–4260.
[2] Barbu, V., R¨ockner, M., From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE. Ann. Probab. 48 (4), 2020, 1902–1920.
[3] Barbu, V., R¨ockner, M., Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations.
J. Funct. Anal. 280 (7), 2021, 108926, 35 p.
[4] Barbu, V., R¨ockner, M., Uniqueness for nonlinear Fokker-Planck equations and weak uniqueness for McKean-Vlasov SDEs, arXiv:1909.04464.
[5] Barbu, V., R¨ockner, M., The evolution to equilibrium of solutions to nonlinear Fokker-Planck equations, arXiv:1904.08291.
Bielefeld University, Germany
E-mail address: [email protected]
THE BOLTZMANN (–ENSKOG) PROCESS
Barbara R¨udiger The theory of SDEs with Poisson noise is used here to identify the “Boltzmann–Enskog-” and “Boltzmann- process”. These are the stochastic processes solving a SDE which corresponding Kolmogorov equation is given by the “Boltzmann–Enskog equation”, resp. “Boltzmann equation”. It turns out that these are the solution of a Mc Kean–Vlasov type SDE with Poisson noise defined through a compensator determined by the density solving the
“Boltzmann–Enskog equation”, resp. “Boltzmann equation”.
The talk is based on joint results with S. Albeverio and P. Sundar, as well as, M. Friesen and P. Sundar.
Bergische Universit ¨at Wuppertal, Germany E-mail address: [email protected]
LIMIT STATES OF MULTI-COMPONENT DISCRETE DYNAMICAL SYSTEMS
O. R. Satur We study models of multicomponent discrete dynamic conflict systems with attractive interaction, which are characterized by a positive value that is called the attractor index. Consider the set of discrete probability measures µi∈ M1+(Ω) on finite space Ω = {ω1, . . . , ωn}, i = 1, m. Each of these measures µi can be identified with a stochastic vector pi= (pij)nj=1, where pij = µi(ωj), i = 1, m, j = 1, n.
The mapping > generates multi-component discrete dynamical systems with trajectories {pt1, pt2, . . . , ptm} −→>,t {pt+11 , pt+12 , . . . , pt+1m }, where the coordinates of each vector pti= (ptij)nj=1are changed according to the equations
pt+1ij = 1
zt ptij(θt+ 1) + τjt, t = 0, 1, . . . . (1) Here θt = θ(pt1, pt2, . . . , pmt ) is a finite positive function, Tt = (τjt)nj=1 is a vector with non-negative coordinates (attractor index), and zt= θt+ 1 + Wtis normalizing denominator, Wt=Pn
j=1τjt. Theorem 1. Let all coordinates of vector wt= (wjt)nj=1, wtj := τ
t j
Wt be bounded and monotonic (increase or decrease independently of each other). Then for all i = 1, m there exist p∞i = limt→∞pti and all limit vectors p∞i coincide with the vector w∞, so their coordinates p∞ij = τ
∞ j
W∞ for all j.
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References
[1] Koshmanenko, V., Kharchenko, N., Fixed points of complex systems with attractive interaction. Methods of Functional Analysis and Topology 23 (2), 2017, 164–176.
[2] Koshmanenko, V. The Spectral Theory of Conflict Dynamical Systems (in Ukrainian). Naukova Dumka, Kyiv, 2016, 287 p.
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska Str. 3, Kyiv 01024, Ukraine E-mail address: [email protected]
ON SKOROKHOD’S METHOD FOR MCKEAN – VLASOV SDE
SA. Yu. Veretennikov For the McKean – Vlasov stochastic equation in Rd
dXt= B[t, Xt, µt]dt + Σ[t, Xt, µt]dWt, t ≥ 0, X0= x0, (1) with
B[t, x, µ] = Z
b(t, x, y)µ(dy), Σ[t, x, µ] = Z
σ(t, x, y)µ(dy) (2)
(the assumption (2) may be generalised), new weak existence and weak uniqueness results are established under relaxed regularity conditions. Weak existence requires a non-degeneracy of diffusion and no more than a linear growth of both coefficients in the state variable.
Here W is a standard d1-dimensional Wiener process, b and σ are vector and matrix Borel functions of corresponding dimensions d and d × d1, µt is the distribution of the process X at time t. The initial data x0 may be random but independent of W .
The talk is based on the joint paper with Yulia Mishura [1]. The method uses Skorokhod’s convergence on a unique probability space and Krylov’s bounds.
Theorem 1. Let E|x0|4< ∞, sups,y(|b(s, x, y)| + kσ(s, x, y)k) ≤ C(1 + |x|), ∀ x, and for any probability measure µ, infs,x inf
|λ|=1λ∗
Z
σ(s, x, y)µ(dy)
Z
σ∗(s, x, y)µ(dy)
λ ≥ ν > 0.
Then the equation (1) has a weak solution on some probability space.
References
[1] Mishura, Yu. S., Veretennikov, A. Yu., Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, https://arxiv.org/abs/1603.02212.
Institute for Information Transmission Problems, B. Karetnii Per. 19, 127051, Moscow, Russian Federation E-mail address: [email protected]
This abstract was prepared within the framework of the HSE University Basic Research Program & the author was supported by Russian Science Foundation grant 17-11-0198.
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EV OLU T ION SY ST EM S, BRAN CHIN G P ROCESSES AN D REN EW AL T HEORY
Dedicated to the main research topics of Volodymyr Korolyuk Chairs A. Iksanov, N. Limnios, I. Samoilenko
VOLODYMYR SEMENOVYCH KOROLYUK (19.08.1925 – 4.04.2020)
Last year Volodymyr Semenovych Korolyuk, a prominent Ukrainian mathematician, one of the founders of the Ukrainian school of probability theory, went to eternity.
V. S. Korolyuk was born in 1925 in Kyiv, where he received a secondary education. Being in the military service, he graduated from the first two courses of Kharkiv University by correspondence, and since 1947 continued his studies at Taras Shechenko Kyiv University, which he graduated in 1950. Research interests of the future scientist were shaped by the influence of Academician B. V. Gnedenko.
Since 1954 he was constantly working at the Institute of Mathematics of NAS of Ukraine (former Academy of Sciences of USSR), first as a research assistant, then since 1956 as a senior researcher, and since 1960 as the Head of the Department of probability theory and mathematical statistics. In 1963 he defended his Doctoral thesis “Asymptotical analysis in boundary problems of random walks”. Since 1966 till 1988 V. S. Korolyuk was the Deputy Director of the Institute of Mathematics of Academy of Sciences of USSR. In 1967 he was elected a Corresponding member and in 1976 the Academician of the Academy of Sciences of USSR.
V. S. Korolyuk was one of the first scientists in Ukraine, who assessed the theoretical and practical importance of semi-Markov processes and attracted the attention of his studentsto this topic. The results of these studies launched a new direction, the theory of asymptotic phase merging and averaging of random processes. They were summarized in the monographs of V. S. Korolyuk and A. F. Turbin: “Semi-Markov processes and their applications” (in Russian 1976),
“Mathematical foundations of the state lumping of large systems” (in Russian 1978, in English 1993) and a manual
“State lumping of large systems” (in Russian 1978).
In the 80s of the last century V. S. Korolyuk launched another new direction, asymptotic analysis of random evolutions. The research results of this direction are summarized in the monographs “Stochastic models of systems”
(in Russian 1989, in Ukrainian 1993, in English 1999, with V. V. Korolyuk as a co-author) and “Semi-Markov random evolutions” with A. V. Swishchuk as a co-author (in Russian 1992, in English 1995).
Since 1990th years V. S. Korolyuk continued study of the new asymptotic methods for evolutionary systems with random perturbations. Long years of creative collaboration in the studying of phase merging of V. S. Korolyuk and N. Limnios, Professor of Technological University of Compiegne (France), promoted in 2005 (the 80th anniversary of Korolyuk) another monograph: V. S. Korolyuk, N. Limnios “Stochastic systems in merging phase space”, published by World Scientific Publishers.
The mathematical heritage of V. S. Korolyuk covers 22 monographs and about 20 textbooks, most of which are reprinted in foreign languages; about 280 scientific articles, about 50 popular science articles and editorial publications to “Encyclopedia of Cybernetics” (in Russian and Ukrainian), monographs, reference books and scientific collections.
The outstanding scientist combined fruitful scientific work, teaching and scientific-organizational activities. Since 1954 he lectured on the theory of programming, probability theory and mathematical statistics at the Taras Shevchenko Kyiv University (Faculty of mechanics and mathematics). The “Manual on probability theory and mathematical statistics” (in Russian) was published in 1978 by his editorship and was repeatedly reprinted in different languages. As
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