Russo F., Vallois P. Stochastic Calculus via Regularizations

Cham: Springer, 2022. — 656 p.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 rely 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 nonpathwise integrals such as Young, fractional, and 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 modeling vortex filaments in turbulence. The book is addressed to Ph.D. students and researchers in stochastic analysis and applications to various fields.About the BookReview of Basic Probability Theory.
Probability Spaces.
The Probability Distribution of a Random Variable.
Expectations of a RV.
Stochastic Independence.
Inequalities and Lp Spaces.
Random Vectors.
Real Gaussian Random Variables.
Gaussian Vectors.
Convergence of a Sequence of RV's.
Limit Theorems.
Conditional Expectation.
Uniform Integrability.
Topological Tools.
A Maximal Inequality.
Processes, Brownian Motion, and Martingales.
Generalities on Continuous Time Processes.
Filtrations and Stopping Times.
Gaussian Random Functions and Processes.
Brownian Motion.
Some Constructions of Brownian Motion.
White Noise.
Continuous Time Martingales.
Local Martingales and Semimartingales.
Fractional Brownian Motion and Related Processes.
Preliminary Considerations.
Fractional Brownian Motion.
Fundamental Martingales Associated with the Fractional Brownian Motion.
Bifractional Brownian Motion.
Stochastic Integration via Regularization.Definitions and Fundamental Properties.
Connections with the p-Variation Concept.
Young Integral in a Simplified Framework.
Introduction to Fractional Calculus.
Fractional Integration.
Functional Spaces Associated with Fractional Brownian motion.
Toward Integration concerning Cadlag Integrators.
An Approach via Integrand Convolution.
Itô Integrals.
The Construction of Itô Integral.
Connections with Calculus via Regularizations.
The Semimartingale Case.
The Brownian Case.
Comparison with the Discretization Approach.
Almost Sure Definition of Stochastic Integrals.
Stability of the Covariation and Itô's Formula.
Stability of the Covariation.
Formulae for Finite Quadratic Variation Processes.
Applications to Semimartingales and Itô Processes.
A Glance to Stochastic Differential Equations.
Applications to Multidimensional Semimartingales and Itô Processes.
An Itô Chain Rule.
About Lévy Area.
Change of Probability and Martingale Representation.Equivalent Probabilities.
Girsanov's Theorem and Exponential Martingales.
Representation of Brownian Martingales.
Girsanov's Formula Related to Fractional Brownian Motion.
About Finite Quadratic Variation: Examples.
General Considerations.
The Föllmer-Wu-Yor Process.
Quadratic Variation of a Gaussian Process.
The α-Variation of Fractional Brownian Motion.
Quadratic Variation of Gaussian Volterra Type Processes.
Processes with a Covariance Measure Structure.
Examples of Processes Having a Covariance Measure.
Hermite Polynomials and Wiener Chaos.
Generalities.
Hermite Polynomials and Local Martingales.
Hermite Polynomials in the Gaussian Case.
Multiple Wiener Integrals.
Iterated Wiener Integrals.
Elements of Wiener Analysis.
The Derivative Operator.
The Divergence Operator.
Link to Stochastic Integrals via Regularization.
Quadratic Variation of a Skorohod Integral.
Malliavin and Wiener Chaos Decomposition.
Elements of Non-causal Calculus.
Preliminaries.
Enlargement of Filtrations.
Substitution Formulae.
Itô Classical Stochastic Differential Equations.
Preliminaries.
Existence and Uniqueness in the Lipschitz Case.
Vector-Valued Stochastic Differential Equations.
Path-Dependent SDEs with Lipschitz Coefficients.
Anticipating SDEs of Forward Type.
Markov Processes and Diffusions.
Flow and Semigroup Associated with a Stochastic Differential Equation.
Infinitesimal Generator of a Diffusion.
Links Between Some Parabolic PDEs and SDEs.
Links Between Some Elliptic PDEs and SDEs.
Itô SDEs with Non-Lipschitz Coefficients.
Generalities.
Existence and Uniqueness in Law.
Existence and Uniqueness in Law: The One-Dimensional Case.
Issues Related to Possible Explosion.
Results on Pathwise Uniqueness.
Bessel Processes.
Time Reversal of Diffusions.
Föllmer – Dirichlet Processes.
Generalities.
Itô Formula Under Weak Smoothness Assumptions.
Bouleau – Yor Formula.
Lyons – Zheng Processes.
Example: Bessel Processes with Positive Dimension.
Application to Fractional Processes.
Weak Dirichlet Processes.
Preliminaries.
Stability Properties.
Volterra Processes and Weak Dirichlet Property.
Weak Dirichlet Processes and Martingale Representation.
Semimartingales and Convolution.
Stochastic Calculus with n-Covariations.
Preliminary Considerations.
Definitions, Notations, and Basic Calculus.
Finite Cubic Variation Processes.
m-Order Type Integrals and Itô Formula.
m-Order ν-Integrals and Related Itô Formula.
Calculus via Regularization and Rough Paths.
Preliminary Notions.
Stochastically Controlled Paths and Gubinelli Derivative.
The Second Order Process and Rough Stochastic Integration.
Rough Stochastic Integration via Regularizations.