Stochastic Calculus via Regularizations

Description

The book constitutes an introduction to stochastic calculus, stochastic differential equations and related topics such as Malliavin calculus. On the other hand it focuses on the techniques of stochastic integration and calculus via regularization initiated by the authors. The definitions relies on a smoothing procedure  of the integrator process, they generalize the usual Itô and Stratonovich integrals for Brownian motion but the integrator could also not be a semimartingale and the integrand is allowed to be anticipating. The resulting calculus requires a simple formalism: nevertheless it entails pathwise techniques even though it takes into account randomness.  It allows connecting different types of pathwise and non pathwise integrals such as Young, fractional, Skorohod integrals, enlargement of filtration and rough paths. The covariation, but also high order variations, play a fundamental role in the calculus via regularization, which can also be applied for irregular integrators. A large class of Gaussian processes, various generalizations of semimartingales such that Dirichlet and weak Dirichlet processes are revisited. Stochastic calculus via regularization has been successfully used in applications, for instance in robust finance and on modeling vortex filaments in turbulence. The book is addressed to PhD students and researchers in stochastic analysis and applications to various fields. Preface About the Book Contents About the Authors 1 Review on Basic Probability Theory 1.1 Probability Spaces 1.2 The Probability Distribution of a Random Variable 1.3 Expectation of a r.v. 1.4 Stochastic Independence 1.5 Inequalities and Lp Spaces 1.6 Random Vectors 1.7 Real Gaussian Random Variables 1.8 Gaussian Vectors 1.9 Convergence of a Sequence of r.v.'s 1.10 Limit Theorems 1.11 Conditional Expectation 1.12 Uniform Integrability 1.13 Topological Tools 1.14 A Maximal Inequality 2 Processes, Brownian Motion and Martingales 2.1 Generalities on Continuous Time Processes 2.2 Filtrations and Stopping Times 2.3 Gaussian Random Functions and Processes 2.4 Brownian Motion 2.5 Some Constructions of Brownian Motion 2.6 White Noise 2.7 Continuous Time Martingales 2.8 Local Martingales and Semimartingales 3 Fractional Brownian Motion and Related Processes 3.1 Preliminary Considerations 3.2 Fractional Brownian Motion 3.3 Fundamental Martingales Associated with the Fractional Brownian Motion 3.4 Bifractional Brownian Motion 4 Stochastic Integration via Regularization 4.1 Foreword 4.2 Definitions and Fundamental Properties 4.3 Connections with the p-Variation Concept 4.4 Young Integral in a Simplified Framework 4.5 Introduction to Fractional Calculus 4.6 Fractional Integration 4.7 Functional Spaces Associated with Fractional BrownianMotion 4.8 Toward Integration with Respect to Cadlag Integrators 4.9 An Approach via Integrand Convolution 5 Itô Integrals 5.1 The Construction of Itô Integral 5.2 Connections with Calculus via Regularizations 5.3 The Semimartingale Case 5.4 The Brownian Case 5.5 Comparison with the Discretization Approach 5.6 Almost Sure Definition of Stochastic Integrals 6 Stability of the Covariation and Itô's Formula 6.1 Stability of the Covariation 6.2 Formulae for Finite Quadratic Variation Processes 6.3 Applications to Semimartingales and Itô Processes 6.4 A Glance to Stochastic Differential Equations 6.5 Applications to Multidimensional Semimartingales and Itô Processes 6.6 An Itô Chain Rule 6.7 About Lévy Area 7 Change of Probability and Martingale Representation 7.1 Foreword 7.2 Equivalent Probabilities 7.3 Girsanov's Theorem and Exponential Martingales 7.4 Representation of Brownian Martingales 7.5 Girsanov's Formula Related to Fractional Brownian Motion 8 About Finite Quadratic Variation: Examples 8.1 General Considerations 8.2 The Föllmer-Wu-Yor Process 8.3 Quadratic Variation of a Gaussian Process 8.4 The α-Variation of Fractional Brownian Motion 8.5 Quadratic Variation of Gaussian Volterra Type Processes 8.6 Processes with a Covariance Measure Structure 8.7 Examples of Processes Having a Covariance Measure 9 Hermite Polynomials and Wiener Chaos 9.1 Generalities 9.2 Hermite Polynomials and Local Martingales 9.3 Hermite Polynomials in the Gaussian Case 9.4 Multiple Wiener Integrals 9.5 Iterated Wiener Integrals 10 Elements of Wiener Analysis 10.1 The Derivative Operator 10.2 The Divergence Operator 10.3 Link to Stochastic Integrals via Regularization 10.4 Quadratic Variation of a Skorohod Integral 10.5 Malliavin and Wiener Chaos Decomposition 11 Elements of Non-causal Calculus 11.1 Preliminaries 11.2 Enlargement of Filtrations 11.3 Substitution Formulae 12 Itô Classical Stochastic Differential Equations 12.1 Preliminaries 12.2 Existence and Uniqueness in the Lipschitz Case 12.3 Vector Valued Stochastic Differential Equations 12.4 Path-Dependent SDEs with Lipschitz Coefficients 12.5 Anticipating SDEs of Forward Type 12.6 Markov Processes and Diffusions 12.7 Flow and Semigroup Associated with a Stochastic Differential Equation 12.8 Infinitesimal Generator of a Diffusion 12.9 Links Between Some Parabolic PDEs and SDEs 12.10 Links Between with Some Elliptic PDEs and SDEs 13 Itô SDEs with Non-Lipschitz Coefficients 13.1 Generalities 13.2 Existence and Uniqueness in Law 13.3 Existence and Uniqueness in Law: The One-Dimensional Case 13.4 Issues Related to Possible Explosion 13.5 Results on Pathwise Uniqueness 13.6 Bessel Processes 13.7 Time Reversal of Diffusions 14 Föllmer–Dirichlet Processes 14.1 Generalities 14.2 Itô Formula Under Weak Smoothness Assumptions 14.3 Bouleau–Yor Formula 14.4 Lyons–Zheng Processes 14.5 Example: Bessel Processes with Positive Dimension 14.6 Application to Fractional Processes 15 Weak Dirichlet Processes 15.1 Preliminaries 15.2 Stability Properties 15.3 Volterra Processes and Weak Dirichlet Property 15.4 Weak Dirichlet Processes and Martingale Representation 15.5 Semimartingales and Convolution 16 Stochastic Calculus with n-Covariations 16.1 Preliminary Considerations 16.2 Definitions, Notations, and Basic Calculus 16.3 Finite Cubic Variation Processes 16.4 m-Order Type Integrals and Itô Formula 16.5 m-Order ν-Integrals and Related Itô Formula 17 Calculus via Regularization and Rough Paths 17.1 Preliminary Notions 17.2 Stochastically Controlled Paths and Gubinelli Derivative 17.3 The Second Order Process and Rough Stochastic Integration 17.4 Rough Stochastic Integration via Regularizations References Index