This book is devoted to stochastic operators in Hilbert space. A number of models in modern probability theory apply the notion of a stochastic operator in explicit or latent form. In this book, objects from the Gaussian case are considered. Therefore, it is useful to consider all random variables and elements as functionals from the Wiener process or its formal derivative, i.e. white noise. The book consists of five chapters. The first chapter is devoted to stochastic calculus and its main goal is to prepare the tools for solving stochastic equations. In the second chapter the structure of stochastic equations, mainly the structure of Gaussian strong linear operators, is studied. In chapter 3 the definition of the action of the stochastic operator on random elements in considered. Chapter 4 deals with the mathematical models in which the notions of stochastic calculus arise and in the final chapter the equation with random operators is considered.

Self-similar processes are stochastic processes that are invariant in distribution under suitable time scaling, and are a subject intensively studied in the last few decades. This book presents the basic properties of these processes and focuses on the study of their variation using stochastic analysis. While self-similar processes, and especially fractional Brownian motion, have been discussed in several books, some new classes have recently emerged in the scientific literature. Some of them are extensions of fractional Brownian motion (bifractional Brownian motion, subtractional Brownian motion, Hermite processes), while others are solutions to the partial differential equations driven by fractional noises. In this monograph the author discusses the basic properties of these new classes of self-similar processes and their interrelationship. At the same time a new approach (based on stochastic calculus, especially Malliavin calculus) to studying the behavior of the variations of self-similar processes has been developed over the last decade. This work surveys these recent techniques and findings on limit theorems and Malliavin calculus.

Chapters 6-9 present methods for solving problems defined by equations with deterministic and/or random coefficients and deterministic and/or stochastic inputs. The Monte Carlo simulation is used extensively throughout to clarify advanced theoretical concepts and provide solutions to a broad range of stochastic problems.".

This volume contains a large spectrum of work: super processes, Dirichlet forms, anticipative stochastic calculus, random fields and Wiener space analysis. The first part of the volume consists of two main lectures given at the third Silivri meeting in 1990: 1. "Infinitely divisible random measures and superprocesses" by D.A. Dawson, 2. "Dirichlet forms on infinite dimensional spaces and appli cations" by M. Rockner. The second part consists of recent research papers all related to Stochastic Analysis, motivated by stochastic partial differ ential equations, Markov fields, the Malliavin calculus and the Feynman path integrals. We would herewith like to thank the ENST for its material support for the above mentioned meeting as well as for the ini tial preparation of this volume and to our friend and colleague Erhan Qmlar whose help and encouragement for the realization of this volume have been essential. H. Korezlioglu A.S. Ustiinel INFINITELY DIVISIBLE RANDOM MEASURES AND SUPERPROCESSES DONALD A. DAWSON 1. Introduction.

This volume examines the theory of fractional Brownian motion and other long-memory processes. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. It proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market.

Problems and Solutions in Stochastic Calculus with Applications exposes readers to simple ideas and proofs in stochastic calculus and its applications. It is intended as a companion to the successful original title Introduction to Stochastic Calculus with Applications (Third Edition) by Fima Klebaner. The current book is authored by three active researchers in the fields of probability, stochastic processes, and their applications in financial mathematics, mathematical biology, and more. The book features problems rooted in their ongoing research. Mathematical finance and biology feature pre-eminently, but the ideas and techniques can equally apply to fields such as engineering and economics.The problems set forth are accessible to students new to the subject, with most of the problems and their solutions centring on a single idea or technique at a time to enhance the ease of learning. While the majority of problems are relatively straightforward, more complex questions are also set in order to challenge the reader as their understanding grows. The book is suitable for either self-study or for instructors, and there are numerous opportunities to generate fresh problems by modifying those presented, facilitating a deeper grasp of the material.

The main purpose of the book is to present, at a graduate level and in a self-contained way, the most important aspects of the theory of continuous stochastic processes in continuous time and to introduce some of its ramifications such as the theory of semigroups, the Malliavin calculus, and the Lyons' rough paths. This book is intended for students, or even researchers, who wish to learn the basics in a concise but complete and rigorous manner. Several exercises are distributed throughout the text to test the understanding of the reader and each chapter ends with bibliographic comments aimed at those interested in exploring the materials further. Stochastic calculus was developed in the 1950s and the range of its applications is huge and still growing today. Besides being a fundamental component of modern probability theory, domains of applications include but are not limited to: mathematical finance, biology, physics, and engineering sciences. The first part of the text is devoted to the general theory of stochastic processes. The author focuses on the existence and regularity results for processes and on the theory of martingales. This allows him to introduce the Brownian motion quickly and study its most fundamental properties. The second part deals with the study of Markov processes, in particular, diffusions. The author's goal is to stress the connections between these processes and the theory of evolution semigroups. The third part deals with stochastic integrals, stochastic differential equations and Malliavin calculus. In the fourth and final part, the author presents an introduction to the very new theory of rough paths by Terry Lyons.