Skip to main content
main-content

Über dieses Buch

This research monograph brings together, for the first time, the varied literature on Yosida approximations of stochastic differential equations (SDEs) in infinite dimensions and their applications into a single cohesive work. The author provides a clear and systematic introduction to the Yosida approximation method and justifies its power by presenting its applications in some practical topics such as stochastic stability and stochastic optimal control. The theory assimilated spans more than 35 years of mathematics, but is developed slowly and methodically in digestible pieces.

The book begins with a motivational chapter that introduces the reader to several different models that play recurring roles throughout the book as the theory is unfolded, and invites readers from different disciplines to see immediately that the effort required to work through the theory that follows is worthwhile. From there, the author presents the necessary prerequisite material, and then launches the reader into the main discussion of the monograph, namely, Yosida approximations of SDEs, Yosida approximations of SDEs with Poisson jumps, and their applications. Most of the results considered in the main chapters appear for the first time in a book form, and contain illustrative examples on stochastic partial differential equations. The key steps are included in all proofs, especially the various estimates, which help the reader to get a true feel for the theory of Yosida approximations and their use.

This work is intended for researchers and graduate students in mathematics specializing in probability theory and will appeal to numerical analysts, engineers, physicists and practitioners in finance who want to apply the theory of stochastic evolution equations. Since the approach is based mainly in semigroup theory, it is amenable to a wide audience including non-specialists in stochastic processes.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction and Motivating Examples

Abstract
Stochastic differential equations are well known to model stochastic processes observed in the study of dynamic systems arising from many areas of science, engineering, and finance. Existence and uniqueness of mild, strong, relaxed, and weak solutions; stability, stabilizability, and control problems; regularity and continuous dependence on initial values; approximation problems notably of Yosida; among others, of solutions of stochastic differential equations in infinite dimensions have been investigated by several authors, see, for instance, Ahmed [1, 6, 8] Bharucha-Reid [1], Curtain and Pritchard [1], Da Prato [2], Da Prato and Zabczyk [1, 3, 4], Gawarecki and Mandrekar [1], Kotelenez [1], Liu [2], Mandrekar and Rüdiger [1], McKibben [2], and Prévôt and Röckner [1] and the references therein. Yosida approximations play a key role in many of these problems.
T. E. Govindan

Chapter 2. Mathematical Machinery

Abstract
The purpose of this chapter is to introduce the necessary background from the semigroup theory, particularly, the Yosida approximations and their properties, analysis and probability in Banach spaces, including Itô stochastic calculus, stochastic convolution integrals, among others. As pointed out before, no attempt has been made to make the presentation self-contained as there are many excellent books available in the literature.
T. E. Govindan

Chapter 3. Yosida Approximations of Stochastic Differential Equations

Abstract
In this chapter, we study Yosida approximations of various classes of stochastic differential equations, including such equations with delays and controlled stochastic differential equations.
T. E. Govindan

Chapter 4. Yosida Approximations of Stochastic Differential Equations with Jumps

Abstract
In this chapter, we consider Yosida approximations of various classes of stochastic differential equations with Poisson jumps.
T. E. Govindan

Chapter 5. Applications to Stochastic Stability

Abstract
In this chapter, we apply some of the results obtained in Chapters 3 and 4 to many problems in stochastic stability. Note that Yosida approximations play a crucial role in these applications.
T. E. Govindan

Chapter 6. Applications to Stochastic Optimal Control

Abstract
In the last chapter of the book, several applications of Yosida approximations are considered for stochastic optimal control problems.
T. E. Govindan

Backmatter

Weitere Informationen

Premium Partner

    Bildnachweise