The Splendors and Miseries of Martingales: Their History from the Casino to Mathematics PDF
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Trends in the History of Science Laurent Mazliak Glenn Shafer Editors The Splendors and Miseries of Martingales Their History from the Casino to Mathematics Trends in the History of Science Trends in the History of Scienceisaseriesdevotedtothepublicationofvolumes arising from workshops and conferences in all areas of current research in the history of science, primarily with a focus on the history of mathematics, physics, and their applications. Its aim is to make current developments available to the community as rapidly as possible without compromising quality, and to archive thosedevelopmentsforreferencepurposes.Proposalsforvolumescanbesubmitted using the online book project submission form at our website www.birkhauser-sci ence.com. Laurent Mazliak · Glenn Shafer Editors The Splendors and Miseries of Martingales Their History from the Casino to Mathematics Editors LaurentMazliak GlennShafer SorbonneUniversité RutgersBusinessSchool LPSM Newark,NJ,USA Paris,France ISSN2297-2951 ISSN2297-296X (electronic) TrendsintheHistoryofScience ISBN978-3-031-05987-2 ISBN978-3-031-05988-9 (eBook) https://doi.org/10.1007/978-3-031-05988-9 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Introduction The mathematical concept of a martingale appears today as one of the essential tools of modern probability theory. Formalized only at the end of the 1930s, even though we can now see it in the pioneering seventeenth and eighteenth centuries work of Blaise Pascal and Abraham De Moivre, the concept gives the discipline an efficient way to obtain a myriad of fundamental results through the relatively elementary verification of a property directly based on the notion of conditional expectation.Asthenamefromtheworldofgamblingindicates,martingalescame into mathematics in the 1930s as strategies for betting. The central mathemati- cal preoccupation of the mathematician who first promoted the name, Jean Ville, was with the asymptotic properties of the evolution of the player’s capital. But the extension of the concept to continuous time brought into view the martingale property of certain random processes, especially the two most important ones, Brownian motion and the Poisson process. With the help of this property, a new type of integral was defined, the stochastic integral, and beyond that a brilliantly original differential calculus, whose results have not ceased to grow in impor- tancesincethemiddleofthetwentiethcentury.Whenwealsobringintoviewthe connections between discrete- or continuous-time Markov processes and various properties of martingales, we see that by the end of the last century martingales had invaded all aspects of probability theory and its applications. v vi Introduction The literature on the history of probability has not kept pace with this growing importance of martingales. The special issue of the ElectronicJournalforHistory ofProbabilityandStatistics that we devoted to the history of martingales in 2009 was, to our knowledge, the first attempt to gather texts and documents that traced with some precision the genesis and the trajectory of the concept through the history of the mathematics of randomness in the twentieth century. The present work can be seen as a considerably enriched second edition of this issue of the electronic journal. This seemed to us a necessary initiative for two reasons. The first is the cessation of the publication of the e-journal, which unfortunately did not survive the death of its co-founder Marc Barbut in 2011: the fragility inherent to the perennial availability of a discontinued online journal made us think that it wouldbejudicioustoguaranteethisavailabilitythroughabookpublishedbothin paper and online. The second reason is more profound: in the more than 10 years thathavepassedsince2009,newresearchhasextended,corrected,andcompleted many of the texts that we presented in 2009, and newly discovered documents and newly emerging insights have added important elements to the puzzle that a historiographic construction always constitutes. As a result, most of the texts presented in this book are either entirely new or significantly enhanced. The book has four principal parts, ordered more or less chronologically. A fifth part presents annotated transcriptions of archival documents that enrich the historical account. The first part of the book, entitled In the beginning, considers some aspects of “martingales before martingales”. In a text full of verve and literary erudition, RogerMansuytracesthegenealogyofthename.Hetellsusthatamartingalecan beapartofahorse’sharness,apartofasailboat’srigging,aman’scoat,orevena courtesan.Lexicographershaveadvancedvarioushypothesesfortheword’sorigin. Mansuybeginswithitsuseinmathematicsandworksbacktothepicturesquecity ofMartigues,ontheFrenchMediterraneancoast.Alongthewayoneencountersa king’s breeches, fencers, a sailor’s dance, a prophetess, and an imprisoned Javert in Victor Hugo’s LesMisérables. The second chapter, by Glenn Shafer, focuses on martingales in games of chance.Shaferaskswhatmadeabettingstrategyamartingaleandwhatmademar- tingales so deceptively alluring to casino gamblers. The decade after the French revolution marks the beginning of a written record that casts some light on these questions. By the beginning of the nineteenth century, as Shafer shows, any bet- ting strategy could be called a martingale. The casino history has more than an antiquarian interest, for we still live with the seduction of martingales, inside the casinoandoutside—whetheratthehorsetrack,inhighfinance,orathomeasday traders and internet gamblers. ThethirdchapterisaprofoundstudybyBernardandMarie-FranceBru,which tries to locate the magic of martingales in a paradox that appeared very early in the history of probability: a fair game can become unfair at infinity. This para- dox was fully understood and mastered only in the 1940s by Émile Borel, who resolveditusingthetheoryofdenumerableprobabilityhehadintroducedin1909. Borel’s reflections on the paradoxes of infinite play were stimulated by a debate Introduction vii beginning around 1910 with the biologist and uncompromising determinist Félix Le Dantec. Despite Le Dantec’s fierce contention that probabilities are of no use inscience, Borel learnedsomething profound and far-reachingabout probability’s applications from him: they generally depend on equating a small or zero prob- ability with impossibility. He struggled for decades to reconcile this insight with his denumerable probabilities. The fourth and final chapter of the first part, by Salah Eid, focuses on the exchanges between the Danish analyst Børge Jessen and the French probabilist Paul Lévy, who arrived from their different starting points at convergent insights that became central to martingale theory. Their main results have come to be known as Jessen’s theorem and Lévy’s lemma. Jessen saw his theorem as an extension of the Fubini-Lebesgue theorem of 1907–1920. Lévy saw his lemma asanextensionofBorel’sstronglawoflargenumbersof1909.Inlettersbetween the two authors, each wanted to see the other’s result as a trivial consequence of their own. Jessen sought a level of abstraction that proved unattainable, but his interaction with Lévy can be seen as the origin of a now-standard version of the martingale convergence theorem. ThesecondpartisentitledVille,Lévy,andDoob,asitfocusesonthethreeprin- cipal protagonists in the emergence of the mathematical concept of a martingale: JeanVille,PaulLévy,andJosephDoob.Thoughhemaybeconsideredchronolog- icallythesecondinline,wehaveplacedJeanVilleastheheroofthefirstchapter of this part, because he was certainly the one who baptized the concept with the name “martingale”. In this chapter, Glenn Shafer enlarges the question, asking what led Ville to think about probability in terms of betting games and explain- inghowbettinggamesallowedhimtolinkBorel’sdenumerableprobabilitieswith Richard von Mises’s concept of probability as frequency in a collective. Shafer seesVilleasexcavatingthemartingalesalreadyhiddeninprobabilitytheory,fore- shadowing their role not only in Lévy’s and Doob’s measure theory but also in two complementary theories—algorithmic complexity theory and game-theoretic probability. As already noted, Ville’s martingales were preceded by Lévy’s. But Lévy was focusedonextendingthelawoflargenumbersandothertheoremsaboutsequences of independent random variables to dependent random variables. In the second chapter of the second part, Laurent Mazliak explains how Lévy showed that this extensionwaspossiblewheneachrandomvariablehasmeanofzerogiventhepre- cedingones.Underthiscondition,thesequenceofcumulativesumsisamartingale as Jean Ville would define the word, but Lévy never focused on this sequence of cumulative sums as a mathematical object. In this respect, his was not a theory of martingales. Moreover, he never showed much interest in the properties of mar- tingales studied by Ville and Doob. In fact, as Mazliak documents, Lévy had a troubled relationship with Ville and generally disdained his mathematical work. Thethirdmathematicianofthefoundingtriad,JosephLeoDoob,putstochastic processes into Kolmogorov’s framework, systematically and with brilliant suc- cess. In the 1940s, he developed a theory of martingales that eclipsed what Lévy and Ville had contributed to the concept. In the third chapter of the second part, viii Introduction by Bernard Locker, provides penetrating insights into Doob’s fundamental con- tribution from the vantage point of his lecture on applications of the theory of martingales at the international colloquium on probability organized in Lyon in 1948 by Maurice Fréchet. It was here, at Lyon, that Doob first used the word “martingale” systematically in his own work, that this work was first presented in France, and that the mathematical world first saw how easily the concept of a martingaleyieldsthestronglawoflargenumbersandthealmostsureconsistency of Bayesian estimation. The third part entitled Modern probability, recounts how martingales came to play so central a role in this modern mathematical theory. Its three chapters are providedbywitnesseswhohadanimportantroleintheevolutionofthefieldafter the Second World War. The first chapter is by Paul-André Meyer, who was the architect of the general theory of processes in the 1960s and 1970s. This theory forms the basis of all subsequent studies using continuous time processes more general than Brownian motion or Poisson processes. The chapter is a translation ofatextwrittenbyMeyerontheeveoftheyear2000,whichtraceshisperception of the evolution of the theory over half a century, an evolution which, of course, involvesmuchmorethantheconceptofmartingales.Meyeremphasizesthefound- ing role of Doob’s StochasticProcesses, published in 1953, which presented tools and topics that fueled probabilistic research for the rest of the century: filtrations, stoppingtimes,martingales,Markovprocesses,diffusions,Itô’sstochasticintegral, andstochasticdifferentialequations.Theperiodfrom1950to1965wasdominated by the study of Markov processes and their connections with potential theory and martingales.Intheperiodfrom1965to1980,martingalesbecamemoreprominent, alongwiththestochasticintegral,excursions,thegeneraltheoryofprocesses,and stochasticmechanics.Thereviewextendsintothe1980s,discussingtheMalliavin calculus and noncommutative probability theory. The second chapter of the third part, provided by another important actor of the period, Shinzo Watanabe, focuses on the contributions of the vigorous and productive Japanese school of probability. Though Japanese scholars did not con- tributedirectlytomartingaletheorybefore1960,manyoftheircontributionsafter 1960 were based on the stochastic calculus that Kiyosi Itô first introduced in 1942. Itô’s collaboration with Henry McKean on the pathwise construction of diffusions attracted wide interest from students in Japan. Subsequent Japanese contributions in the 1960s included adaptations of results on Markov processes to martingales, such as Itô and Watanabe’s multiplicative analog of the Doob-Meyer decomposition,whichinvolvedtheintroductionoflocalmartingales,contributions to stochastic integration for square-integrable martingales and semi-martingales, andcontributionstotherepresentationofmartingales.Japanesecontributionsafter 1970 included Itô’s reformulation of the stochastic calculus in terms of stochas- tic differentials, Itô’s circle operation, the Itô-Tanaka formula, and the Fukushima decomposition. The third chapter of the third part is an autobiographical account by Klaus Krickeberg. As a university student at Humboldt University in the difficult con- ditions right after the Second World War, he was attracted to mathematics by the Introduction ix brilliantteachingofthefamousanalystErhardSchmidt.AftermovingfromBerlin toWürzburgin1953,hebecameacquaintedwithDoob’sworkonmartingalesand Dieudonné’scounterexampletoDoob’smartingaleconvergencetheorem.Thisled to his work on the role of Vitali-type conditions in the convergence. After obtain- inghisHabilitationinWürzburg,hespentayearinDoob’sgroupattheUniversity of Illinois. During this period, he proved that every L1-bounded increasing semi- martingale with a directed index set converges stochastically. His further work on martingales at Würzburg from 1957 to 1964 continued to emphasize Vitali conditions. Thefourthpart,entitledModernapplications,reviewsafewoftheapplications of martingale theory. Its first chapter, by Laurent Bienvenu, Glenn Shafer and Alexander Shen, recounts the role played by the concept of a martingale in the algorithmic understanding of randomness. In the 1930s, Jean Ville used martin- gales to improve Richard von Mises’s and Abraham Wald’s concept of an infinite randomsequence,orcollective.Afterthedevelopmentofalgorithmicrandomness by Andrei Kolmogorov, Ray Solomonoff, Gregory Chaitin, and Per Martin-Löf in the 1960s, Claus-Peter Schnorr developed Ville’s concept in this new context. Along with Schnorr, Leonid Levin was a key figure in the development in the 1970s. While Schnorr worked with algorithmic martingales and supermartingales, Levin worked with the closely related concept of a semi-measure. In order to characterize the randomness of an infinite sequence in terms of the complexity of its prefixes, they introduced new ways of measuring complexity: monotone complexity (Schnorr and Levin) and prefix complexity (Levin and Chaitin). ThesecondchapterinthefourthpartprovidedbyTzeLeungLai,describeshow martingales came into his world of mathematical statistics, first in sequential tests and confidence intervals, then in time series, stochastic approximation, sequential design of experiments, and stochastic optimization. Lai sketches the trajectories of many other statisticians that he met along the way, emphasizing the roles of Harold Hotelling, Abraham Wald, and Herbert Robbins in their creation of the environmentforthestudyofmartingalesatColumbiaUniversityandthenhisown subsequent work at Stanford University. At Stanford, he came to see stochastic optimizationasaunifyingthemefortheuseofmartingalesinstatisticalmodeling. Another applied field where martingales have made a great contribution is sur- vival analysis. In their chapter, Odd O. Aalen, Per K. Andersen, Ørnulf Borgan, RichardD.Gill,andNielsKeidingtracethedevelopmentofmartingalesinsurvival analysis from the mid-1970s to the early 1990s. This development was initiated by Aalen’s Berkeley Ph.D. thesis in 1975, progressed in the late 1970s and early 1980s through work on the estimation of Markov transition probabilities, non- parametric tests, and Cox’s regression model, and was consolidated in the early 1990s with the publication of the monographs by Fleming and Harrington and by Andersen, Borgan, Gill, and Keiding. The authors see this development as an unusually fast technology transfer of pure mathematical concepts, primarily from French probability, into a practical biostatistical methodology. It was possi- ble because the martingale ideas inherent in the deep understanding of temporal
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