Table Of Content(cid:2)
Game-Theoretic
Foundations for Probability
and Finance
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Game-Theoretic
Foundations for Probability
and Finance
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GLENNSHAFER
RutgersBusinessSchool
VLADIMIRVOVK
RoyalHolloway,UniversityofLondon
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Thiseditionfirstpublished2019
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LibraryofCongressCataloging-in-PublicationData
Names:Shafer,Glenn,1946-author.|Vovk,Vladimir,1960-author.
Title:Game-theoreticfoundationsforprobabilityandfinance/GlennRay
Shafer,RutgersUniversity,NewJersey,USA,VladimirVovk,Universityof
London,Surrey,UK.
Othertitles:Probabilityandfinance
Description:Firstedition.|Hoboken,NJ:JohnWiley&Sons,Inc.,2019.|
Series:Wileyseriesinprobabilityandstatistics|Earlieredition
publishedin2001as:Probabilityandfinance:it’sonlyagame!|
Includesbibliographicalreferencesandindex.|
Identifiers:LCCN2019003689(print)|LCCN2019005392(ebook)|ISBN
9781118547939(AdobePDF)|ISBN9781118548028(ePub)|ISBN9780470903056
(hardcover)
Subjects:LCSH:Finance–Statisticalmethods.|Finance–Mathematicalmodels.
|Gametheory.
Classification:LCCHG176.5(ebook)|LCCHG176.5.S532019(print)|DDC
332.01/5193–dc23
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10 9 8 7 6 5 4 3 2 1
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Contents
Preface xi
Acknowledgments xv
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PartI ExamplesinDiscreteTime 1
1 Borel’sLawofLargeNumbers 5
1.1 AProtocolforTestingForecasts 6
1.2 AGame-TheoreticGeneralization ofBorel’sTheorem 8
1.3 BinaryOutcomes 16
1.4 SlackeningsandSupermartingales 18
1.5 Calibration 19
1.6 TheComputation ofStrategies 21
1.7 Exercises 21
1.8 Context 24
2 Bernoulli’sandDeMoivre’sTheorems 31
2.1 Game-TheoreticExpectedValueandProbability 33
2.2 Bernoulli’sTheoremforBoundedForecasting 37
2.3 ACentralLimitTheorem 39
2.4 GlobalUpperExpectedValuesforBoundedForecasting 45
v
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vi CONTENTS
2.5 Exercises 46
2.6 Context 49
3 SomeBasicSupermartingales 55
3.1 Kolmogorov’sMartingale 56
3.2 Doléans’sSupermartingale 56
3.3 Hoeffding’sSupermartingale 58
3.4 Bernstein’sSupermartingale 63
3.5 Exercises 66
3.6 Context 67
4 Kolmogorov’sLawofLargeNumbers 69
4.1 Stating Kolmogorov’sLaw 70
4.2 SupermartingaleConvergenceTheorem 73
4.3 HowSkepticForcesConvergence 80
4.4 HowRealityForcesDivergence 81
4.5 ForcingGames 82
4.6 Exercises 86
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4.7 Context 89
5 TheLawoftheIteratedLogarithm 93
5.1 ValidityoftheIterated-Logarithm Bound 94
5.2 SharpnessoftheIterated-LogarithmBound 99
5.3 Additional RecentGame-TheoreticResults 100
5.4 ConnectionswithLargeDeviationInequalities 104
5.5 Exercises 104
5.6 Context 106
PartII AbstractTheoryinDiscreteTime 109
6 BettingonaSingleOutcome 111
6.1 UpperandLowerExpectations 113
6.2 UpperandLowerProbabilities 115
6.3 UpperExpectationswithSmallerDomains 118
6.4 Offers 121
6.5 DroppingtheContinuity Axiom 125
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CONTENTS vii
6.6 Exercises 127
6.7 Context 131
7 AbstractTestingProtocols 135
7.1 TerminologyandNotation 136
7.2 Supermartingales 136
7.3 GlobalUpperExpectedValues 142
7.4 Lindeberg’sCentralLimitTheoremforMartingales 145
7.5 GeneralAbstractTestingProtocols 146
7.6 MakingtheResultsofPartIAbstract 151
7.7 Exercises 153
7.8 Context 155
8 Zero-OneLaws 157
8.1 Lévy’sZero-OneLaw 158
8.2 GlobalUpperExpectation 160
8.3 GlobalUpperandLowerProbabilities 162
8.4 GlobalExpectedValuesandProbabilities 163
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8.5 OtherZero-OneLaws 165
8.6 Exercises 169
8.7 Context 170
9 RelationtoMeasure-TheoreticProbability 175
9.1 Ville’sTheorem 176
9.2 Measure-TheoreticRepresentationofUpper
Expectations 180
9.3 EmbeddingGame-TheoreticMartingalesinProbability
Spaces 189
9.4 Exercises 191
9.5 Context 192
PartIII ApplicationsinDiscreteTime 195
10 UsingTestingProtocolsinScienceandTechnology 197
10.1 SignalsinOpenProtocols 198
10.2 Cournot’sPrinciple 201
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viii CONTENTS
10.3 Daltonism 202
10.4 LeastSquares 207
10.5 ParametricStatistics withSignals 212
10.6 QuantumMechanics 215
10.7 Jeffreys’sLaw 217
10.8 Exercises 225
10.9 Context 226
11 Calibrating Lookbacksandp-Values 229
11.1 LookbackCalibrators 230
11.2 LookbackProtocols 235
11.3 LookbackCompromises 241
11.4 LookbacksinFinancial Markets 242
11.5 Calibratingp-Values 245
11.6 Exercises 248
11.7 Context 250
12 DefensiveForecasting 253
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12.1 DefeatingStrategiesforSkeptic 255
12.2 CalibratedForecasts 259
12.3 ProvingtheCalibration Theorems 264
12.4 UsingCalibrated ForecastsforDecisionMaking 270
12.5 ProvingtheDecisionTheorems 274
12.6 FromTheorytoAlgorithm 286
12.7 DiscontinuousStrategiesforSkeptic 291
12.8 Exercises 295
12.9 Context 299
PartIV Game-TheoreticFinance 305
13 EmergenceofRandomnessinIdealizedFinancialMarkets 309
13.1 CapitalProcessesandInstant Enforcement 310
13.2 EmergenceofBrownianRandomness 312
13.3 EmergenceofBrownianExpectation 320
13.4 ApplicationsofDubins–Schwarz 325
13.5 GettingRichQuickwiththeAxiomofChoice 331
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CONTENTS ix
13.6 Exercises 333
13.7 Context 334
14 AGame-TheoreticItôCalculus 339
14.1 MartingaleSpaces 340
14.2 ConservatismofContinuous Martingales 348
14.3 ItôIntegration 350
14.4 CovariationandQuadraticVariation 355
14.5 Itô’sFormula 357
14.6 DoléansExponentialandLogarithm 358
14.7 Game-TheoreticExpectationandProbability 360
14.8 Game-TheoreticDubins–SchwarzTheorem 361
14.9 Coherence 362
14.10Exercises 363
14.11Context 365
15 NumerairesinMarketSpaces 371
15.1 MarketSpaces 372
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15.2 MartingaleTheoryinMarketSpaces 375
15.3 Girsanov’sTheorem 376
15.4 Exercises 382
15.5 Context 382
16 EquityPremiumandCAPM 385
16.1 ThreeFundamentalContinuousI-Martingales 387
16.2 EquityPremium 389
16.3 CapitalAssetPricingModel 391
16.4 TheoreticalPerformanceDeficit 395
16.5 SharpeRatio 396
16.6 Exercises 397
16.7 Context 398
17 Game-TheoreticPortfolioTheory 403
17.1 Stroock–VaradhanMartingales 405
17.2 BoostingStroock–VaradhanMartingales 407
17.3 OutperformingtheMarketwithDubins–Schwarz 413
17.4 Jeffreys’sLawinFinance 414
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