This book will be of substantial interest to the applied scientists seeking methods to solve stochastic problems. Each chapter ends with a number of problems. Although the focus is on the mathematical theory, many examples from sciences or engineering illustrate the concepts and methods. Its a great way to give a feel for the structure of the subject without needing. Shreve volume 1 builds a good foundation. "This is an impressive compendium, which, on its first 280 pages, provides a quite thorough review of probability theory, stochastic processes, Itô’s formula and stochastic differential equations. Took an intro to stochastic calculus last semester and ended up using this book as a reference a lot. Altogether this textbook has all the necessary prerequisites for boosting the interdisciplinary collaboration between applied scientists and probabilists." -Zentralblatt Math The long list of references covers theoretical mathematical works as well as highly applied areas. At the end of each chapter a list of instructive exercises is provided. Most results are proved completely, for the others the main ideas and intuition are given and references for detailed expositions are given. The presentation is very successful in providing a sound mathematical background without getting lost in too much technical detail. The applications range from climate dynamics over material sciences and mechanics to pattern formation, physics and finance. "The large number of examples.as well as the good survey on advanced mathematical techniques make the textbook very valuable for people working in mathematics with a view towards applications. ![]() The book is an impressive achievement." -Mathematical Reviews "When familiar at least with the basics of measure theoretic probability, one may use this book to get a feel for the type of problems one tackles with the given methods and the type of results one can expect and then proceed to more detailed expositions. It is assumed that the operators and inputs defining a stochastic problem are specified. The main objective of this book is the solution of stochastic problems, that is, the determination of the probability law, moments, and/or other probabilistic properties of the state of a physical, economic, or social system. Problems that can be defined by algebraic, differential, and integral equations with random coefficients and/or input are referred to as stochastic problems. ![]() For example, the orientation of the atomic lattice in the grains of a polycrystal varies randomly from grain to grain, the spa tial distribution of a phase of a composite material is not known precisely for a particular specimen, bone properties needed to develop reliable artificial joints vary significantly with individual and age, forces acting on a plane from takeoff to landing depend in a complex manner on the environmental conditions and flight pattern, and stock prices and their evolution in time depend on a large number of factors that cannot be described by deterministic models. Generally, the coefficients of and/or the input to these equations are not precisely known be cause of insufficient information, limited understanding of some underlying phe nomena, and inherent randonmess. Change of probability measure and drift coefficient for solutions of stochastic differential equations with applications to likelihood principles and statistical inference.Algebraic, differential, and integral equations are used in the applied sciences, en gineering, economics, and the social sciences to characterize the current state of a physical, economic, or social system and forecast its evolution in time. Change of probability measure for stochastic variables. Time homogeneous diffusion processes together with explosion, recurrence, transience and stationary distributions thereof. Kolmogorov¿s equations together with Dynkin¿s formula and the Feynman-Kac formula. ![]() Stochastic exponential, stochastic logarithm and linear stochastic differential equations. Stochastic differential equations together with existence, uniqueness and Markov property of weak solutions and strong solutions thereof. Ito integrals, Ito integral processes and Ito¿s formula. Defining properties of martingales and Markov processes with continuous time and continuous values. Brownian motion (Wiener process) together with its most important properties. Introduction to axiomatic probability theory and to abstract conditional expectation with respect to sigma fields. Review of Riemann integral, Riemann-Stieltjes integral and Lebesgue integral. Content Variation and quadratic variation of functions.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |