Contents. 1 Preliminaries 3. Martingales

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1 Table of Preface PART I THE FUNDAMENTAL PRINCIPLES page xv 1 Preliminaries 3 2 Martingales Martingales and examples Stopping times The maximum inequality Doob s inequality The σ -algebra over the past of a stopping time L p -spaces of martingales and the quadratic variation norm The supremum norm ( Martingales ) of bounded mean oscillation L 1 [ is BMO2 21 ( ) 2.10 L 1 is BMO B D G inequalities for p = The B D G inequalities for the conditional expectation for p = The B D G inequalities The B D G inequalities for special convex functions 32 Exercises 36 3 Fourier and Laplace transformations Transformations of measures Laplace characterization of N (0,σ)-distribution Fourier and Laplace characterization of independence 39 vii

2 Table of viii 3.4 Discrete Lévy processes and their representation Martingale characterization of Brownian motion 45 Exercises 46 4 Abstract Wiener Fréchet spaces Projective systems of measures and their limit Gaussian measures in Hilbert spaces Abstract Wiener spaces Cylinder sets in Fréchet spaces generate the Borel sets Cylinder sets in Fréchet space valued continuous functions Tensor products Bochner integrable functions The Wiener measure on C B is the centred Gaussian measure of variance 1 66 Exercises 70 5 Two concepts of no-anticipation in time Predictability and adaptedness Approximations of the Dirac δ-function Convolutions of adapted functions are adapted Adaptedness is equivalent to predictability The weak approximation property Elementary facts about L p -spaces 78 Exercises 81 6 Malliavin calculus on real sequences Orthogonal polynomials Integration Iterated integrals Chaos decomposition Malliavin derivative and Skorokhod integral The integral as a special case of the Skorokhod integral The Clark Ocone formula Examples 91 Exercises 94 7 Introduction to poly-saturated models of mathematics Models of mathematics The main theorem 102 Exercises 105

3 Table of ix 8 Extension of the real numbers and properties R as an ordered field The extension of the positive integers Hyperfinite sets and summation in R The underspill and overspill principles The infinitesimals Limited and unlimited numbers in R The standard part map on limited numbers 112 Exercises Topology Monads Hausdorff spaces Continuity Compactness Convergence The standard part of an internal set of nearstandard points is compact From S-continuous to continuous functions Hyperfinite representation of the tensor product The Skorokhod topology 124 Exercises Measure and integration on Loeb spaces The construction of Loeb measures Loeb measures over Gaussian measures Loeb measurable functions On Loeb product spaces Lebesgue measure as a counting measure Adapted Loeb spaces S-integrability and equivalent conditions Bochner integrability and S-integrability Integrable functions defined on N n [0, [ m Standard part of the conditional expectation Witnesses of S-integrability Keisler s Fubini theorem S-integrability of internal martingales S-continuity of internal martingales On symmetric functions The standard part of internal martingales 166 Exercises 170

4 Table of x PART II AN INTRODUCTION TO FINITE- AND INFINITE-DIMENSIONAL STOCHASTIC ANALYSIS Introduction From finite- to infinite-dimensional Brownian motion On the underlying probability space The internal Brownian motion S-integrability of the internal Brownian motion The S-continuity of the internal Brownian motion One-dimensional Brownian motion Lévy s inequality The final construction The Wiener space 190 Exercises The Itô integral for infinite-dimensional Brownian motion The S-continuity of the internal Itô integral On the S-square-integrability of the internal Itô integral The standard Itô integral On the integrability of the Itô integral W CH is generated by the Wiener integrals The distribution of the Wiener integrals 209 Exercises The iterated integral The iterated integral with and without parameters The product of an internal iterated integral and an internal Wiener integral The continuity of the standard iterated integral process The W CH -measurability of the iterated Itô integral In M ( f ) is a continuous version of the standard part of In M (F) Continuous versions of internal iterated integral processes Kolmogorov s continuity criterion 224 Exercises 228

5 Table of xi 14 Infinite-dimensional Ornstein Uhlenbeck processes Ornstein Uhlenbeck processes for shifts given by Hilbert Schmidt operators Ornstein Uhlenbeck processes for shifts by scalars 239 Exercises Lindstrøm s construction of standard Lévy processes from discrete ones Exponential moments for processes with limited increments Limited Lévy processes Approximation of limited processes by processes with limited increments Splitting infinitesimals Standard Lévy processes Lévy measure The Lévy Khintchine formula Lévy triplets generate Lévy processes Each Lévy process can be divided into its continuous and pure jump part 266 Exercises Stochastic integration for Lévy processes Orthogonalization of the increments From internal random walks to the standard Lévy integral Iterated integrals Multiple integrals The σ -algebra generated by the Wiener Lévy integrals 283 Exercises 286 PART III MALLIAVIN CALCULUS Introduction Chaos decomposition Admissible sequences Chaos expansion A lifting theorem for functionals in L 2 W ( Ɣ ) Chaos for functions without moments Computation of the kernels 300

6 Table of xii 17.6 The kernels of the product of Wiener functionals 303 Exercises The Malliavin derivative The domain of the derivative The Clark Ocone formula A lifting theorem for the derivative The directional derivative A commutation rule for derivative and limit The domain of the Malliavin derivative is a Hilbert space with respect to the norm 1, The range of the Malliavin derivative is closed A commutation rule for the directional derivative Product and chain rules for the Malliavin derivative 315 Exercises The Skorokhod integral Decomposition of processes Malliavin derivative of processes The domain of the Skorokhod integral A lifting theorem for the integral The Itô integral is a special case of the Skorokhod integral 325 Exercises The interplay between derivative and integral The integral is the adjoint operator of the derivative A Malliavin differentiable function multiplied by square-integrable deterministic functions is Skorokhod integrable The duality between the domains of D and δ L 2 ( Ɣ ν,h) is the orthogonal sum of the range W L 1 of D and the kernel of δ Integration by parts 334 Exercises Skorokhod integral processes The Skorokhod integral process operator On continuous versions of Skorokhod integral processes 336 Exercises 338

7 Table of xiii 22 Girsanov transformations From standard to internal shifts The Jacobian determinant of the internal shift Time-anticipating Girsanov transformations Adapted Girsanov transformation Extension of abstract Wiener spaces 348 Exercises Malliavin calculus for Lévy processes Chaos Malliavin derivative The Clark Ocone formula Skorokhod integral processes Smooth representations A commutation rule for derivative and limit The product rule The chain rule Girsanov transformations 368 Exercises 374 APPENDICES EXISTENCE OF POLY-SATURATED MODELS Appendix A. Poly-saturated models 379 A.1 Weak models and models of mathematics 379 A.2 From weak models to models 380 A.3 Languages for models 381 A.4 Interpretation of the language 382 A.5 Models closed under definition 383 A.6 Elementary embeddings 384 A.7 Poly-saturated models 386 Appendix B. The existence of poly-saturated models 388 B.1 From pre-models to models 388 B.2 Ultrapowers 390 B.3 Elementary chains and their elementary limits 393 B.4 Existence of poly-saturated models with the same properties as standard models 395 References 398 Index 404