Equations Involving Malliavin Calculus Operators

This book provides a comprehensive and unified introduction to stochastic differential equations and related optimal control problems. The material is new and the presentation is reader-friendly. A major contribution of the book is the development of generalized Malliavin calculus in the framework of white noise analysis, based on chaos expansion representation of stochastic processes and its application for solving several classes of stochastic differential equations with singular data involving the main operators of Malliavin calculus. In addition, applications in optimal control and numerical approximations are discussed.

The book is divided into four chapters. The first, entitled White Noise Analysis and Chaos Expansions, includes notation and provides the reader with the theoretical background needed to understand the subsequent chapters. In particular, we introduce spaces of random variables and stochastic processes, and consider processes that have finite variance on classical and fractional Gaussian white noise probability spaces. We also present processes with infinite variance, particularly Kondratiev stochastic distributions. We introduce the Wick and ordinary multiplication of the processes and state where these operations are well defined.

In Chapter 2, Generalized Operators of Malliavin Calculus, the Malliavin derivative operator D, the Skorokhod integral δ and the Ornstein-Uhlenbeck operator R are introduced in terms of chaos expansions. The main properties of the operators, which are known in the literature for the square integrable processes, are proven using the chaos expansion approach and extended for generalized and test stochastic processes. Moreover, we discuss fractional versions of these operators.

Chapter 3, Equations involving Malliavin Calculus operators, is devoted to the study of several types of stochastic differential equations that involve the operators of Malliavin calculus, introduced in the previous chapter. In particular, we describe the range of the operators D, δ and R.

Finally, in Chapter 4, Applications and Numerical Approximations are discussed. Specifically, we consider the stochastic linear quadratic optimal control problem with different forms of noise disturbances, operator differential algebraic equations arising in fluid dynamics, stationary equations and fractional versions of the equations studied – applications never covered in the extant literature. Moreover, numerical validations of the method are provided for specific problems.