Stochastic Calculus: Applications in Science and Engineering

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. 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. 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. Problems that can be defined by algebraic, differential, and integral equations with random coefficients and/or input are referred to as stochastic problems. 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. It is assumed that the operators and inputs defining a stochastic problem are specified.

This work focuses on analyzing and presenting solutions for a wide range of stochastic problems that are encountered in applied mathematics, probability, physics, engineering, finance, and economics. The approach used reduces the gap between the mathematical and engineering literature. Stochastic problems are defined by algebraic, differential or integral equations with random coefficients and/or input. However, it is the type, rather than the particular field of application, that is used to categorize these problems.

An introductory chapter outlines the types of stochastic problems under consideration in this book and illustrates some of their applications. A user friendly, systematic exposition unfolds as follows:  The essentials of probability theory, random processes, stochastic integration, and Monte Carlo simulation are developed in chapters 2--5. The Monte Carlo method is used extensively to illustrate difficult theoretical concepts and solve numerically some ofthe stochastic problems in chapters 6--9.

Key features:

* Computational skills developed as needed to solve realistic stochastic problems

* Classical mathematical notation used, and essential theoretical facts boxed

* Numerous examples from applied sciences and engineering

* Complete proofs given; if too technical, notes clarify the idea and/or main steps

* Problems at the end of each chapter reinforce applications; hints given.

* Good bibliography at the end of every chapter

* Comprehensive index

This work is unique, self-contained, and far from a collection of facts and formulas. The analytical and numerical methods approach for solving stochastic problems may be used for self-study by a variety of researchers, and in the classroom by first year graduate students.