Stochastic Differential Equations

The exposition of this Book begins with essential elements of stochastic analysis, stochastic calculus, and elements of functional analysis. We then progress to detailed discussions on existence, uniqueness, and stability of solutions, as well as qualitative behaviors under varying conditions. The text also incorporates selected applications, illustrating how stochastic models naturally arise in diverse scientific and engineering problems. This work is intended for graduate students, researchers, and professionals who wish to deepen their understanding of stochastic systems. It may serve as both a book for beginner researchers and a reference for specialists pursuing further studies in the field. The presentation balances rigor with accessibility, combining mathematical depth with an emphasis on clarity. We are indebted to the contributions of many mathematicians whose pioneering work laid the foundations of this subject. Our gratitude extends to colleagues and students whose questions and insights have helped shape the material presented here.
In the first two chapters, the book introduces selected topics from probability the ory: Brownian motion and the Wiener process, the stochastic integral in Hilbert spaces, and fractional Brownian motion. It explains in detail the essential properties of functional analysis, such as generalized metrics and Banach spaces, compactness criteria, measures of non-compactness (MNC), fixed point theory, some properties of set-valued maps, fixed point results, and semi-group theory. The question of the quantitative study of impulsive stochastic differential equations/ systems is treated with particular attention in Chapter 3 and Chapter 4. With fixed moments and multiple delays, the existence of solutions with fixed moments and multiple delays is addressed through the application of Schaefer and Perov fixed point theorems in generalized Banach spaces, driven by standard Brownian motion. WhereasinChapter5, sufficient conditions for the local and global existence and exponential stability of mild solutions of semi-linear systems of stochastic differential equations with infinite fractional Brownian motions and impulses are established with the Hurst index H > 1/2. In Chapter 6, we discuss some results on the existence and uniqueness of mild solutions for systems of semilinear impulsive differential equations with infinite fractional Brownian motions and Wiener processes. The approach is based on a new ver sion of the fixed point theorem due to Krasnoselskii in generalized Banach spaces. Chapter 7 deals with impulsive neutral stochastic functional differential equations driven by fBm with a noncompact semigroup. In Chapter 8, we prove some existence results based on a nonlinear alternative of the Leray-Schauder type theorem in generalized Banach spaces for the convex case; we establish a multi-valued version of Perov's fixed point theorem in a non-convex setting. In Chapter 9, we provide sufficient conditions for the existence of solutions for a class of second-order systems of stochastic impulsive differential inclusions. In Chapter 10, it is devoted to the study of a convex stochastic sweeping process with fractional Brownian motion and time delay. The approach is based on discretizing stochastic functional differential inclusions.
This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers and biologists. It can be used as a textbook at the graduate level and as a reference book for several disciplines.
The aim of this book is to present a clear and well-organized treatment of the concept behind the development of mathematics as well as solution techniques. The text material of this book is presented in a readable and mathematically solid format.