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1988 | Book

Martingales, Stopping Times, and Filtrations Read chapter Read first chapter

Brownian Motion and Stochastic Calculus

Authors: Ioannis Karatzas, Steven E. Shreve

Publisher: Springer US

Book Series : Graduate Texts in Mathematics

Included in: Professional Book Archive

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About this book

Two of the most fundamental concepts in the theory of stochastic processes are the Markov property and the martingale property. * This book is written for readers who are acquainted with both of these ideas in the discrete-time setting, and who now wish to explore stochastic processes in their continuous­ time context. It has been our goal to write a systematic and thorough exposi­ tion of this subject, leading in many instances to the frontiers of knowledge. At the same time, we have endeavored to keep the mathematical prerequisites as low as possible, namely, knowledge of measure-theoretic probability and some familiarity with discrete-time processes. The vehicle we have chosen for this task is Brownian motion, which we present as the canonical example of both a Markov process and a martingale. We support this point of view by showing how, by means of stochastic integration and random time change, all continuous-path martingales and a multitude of continuous-path Markov processes can be represented in terms of Brownian motion. This approach forces us to leave aside those processes which do not have continuous paths. Thus, the Poisson process is not a primary object of study, although it is developed in Chapter 1 to be used as a tool when we later study passage times and local time of Brownian motion.

Table of Contents

Frontmatter
Chapter 1. Martingales, Stopping Times, and Filtrations
Abstract
A stochastic process is a mathematical model for the occurrence, at each moment after the initial time, of a random phenomenon. The randomness is captured by the introduction of a measurable space (Ω, F) called the sample space, on which probability measures can be placed. Thus, a stochastic process is a collection of random variables X = {X t ; 0 ≤ t ∞} on (Ω, F) which take values in a second measurable space (S, S), called the state space. For our purposes, the state space (S, S) will be the d-dimensional Euclidean space equipped with the σ-field of Borel sets, i.e., S = ℝ d , S = B(ℝ d ) where B(U) will always be used to denote the smallest σ-fîeld containing all open sets of a topological space U. The index t ∈ [0, ∞) of the random variables X t admits a convenient interpretation as time.
Ioannis Karatzas, Steven E. Shreve
Chapter 2. Brownian Motion
Abstract
Brownian movement is the name given to the irregular movement of pollen, suspended in water, observed by the botanist Robert Brown in 1828. This random movement, now attributed to the buffeting of the pollen by water molecules, results in a dispersal or diffusion of the pollen in the water. The range of application of Brownian motion as defined here goes far beyond a study of microscopic particles in suspension and includes modeling of stock prices, of thermal noise in electrical circuits, of certain limiting behavior in queueing and inventory systems, and of random perturbations in a variety of other physical, biological, economic, and management systems. Furthermore, integration with respect to Brownian motion, developed in Chapter 3, gives us a unifying representation for a large class of martingales and diffusion processes. Diffusion processes represented this way exhibit a rich connection with the theory of partial differential equations (Chapter 4 and Section 5.7). In particular, to each such process there corresponds a second-order parabolic equation which governs the transition probabilities of the process.
Ioannis Karatzas, Steven E. Shreve
Chapter 3. Stochastic Integration
Abstract
A tremendous range of problems in the natural, social, and biological sciences came under the dominion of the theory of functions of a real variable when Newton and Leibniz invented the calculus. The primary components of this invention were the use of differentiation to describe rates of change, the use of integration to pass to the limit in approximating sums, and the fundamental theorem of calculus, which relates the two concepts and thereby makes the latter amenable to computation. All of this gave rise to the concept of ordinary differential equations, and it is the application of these equations to the modeling of real-world phenomena which reveals much of the power of calculus.
Ioannis Karatzas, Steven E. Shreve
Chapter 4. Brownian Motion and Partial Differential Equations
Abstract
There is a rich interplay between probability theory and analysis, the study of which goes back at least to Kolmogorov (1931). It is not possible in a few sections to develop this subject systematically; we instead confine our attention to a few illustrative cases of this interplay. Recent monographs on this subject are those of Doob (1984) and Durrett (1984).
Ioannis Karatzas, Steven E. Shreve
Chapter 5. Stochastic Differential Equations
Abstract
We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.
Ioannis Karatzas, Steven E. Shreve
Chapter 6. P. Lévy’s Theory of Brownian Local Time
Abstract
This chapter is an in-depth study of the Brownian local time first encountered in Section 3.6. Our approach to this subject is motivated by the desire to perform computations. This is manifested by the inclusion of the conditional Laplace transform formulas of D. Williams (Subsections 6.3.B, 6.4.C), the derivation of the joint density of Brownian motion, its local time at the origin and its occupation time of the positive half-line (Subsection 6.3.C), and the computation of the transition density for Brownian motion with two-valued drift (Section 6.5). This last computation arises in the problem of controlling the drift of a Brownian motion, within prescribed bounds, so as to keep the controlled process near the origin.
Ioannis Karatzas, Steven E. Shreve
Backmatter
Metadata
Title
Brownian Motion and Stochastic Calculus
Authors
Ioannis Karatzas
Steven E. Shreve
Copyright Year
1988
Publisher
Springer US
Electronic ISBN
978-1-4684-0302-2
Print ISBN
978-1-4684-0304-6
DOI
https://doi.org/10.1007/978-1-4684-0302-2