Brownian Motion.
Abstract
The most fundamental concepts in the theory of stochastic processes are the Markov property and the martingale property. This book is written for those who are familiar with both of these ideas in the discrete-time setting, and who now wish to explore stochastic processes in the continuous-time context. It has been our goal to write a systematic and thorough exposition of this subject, leading in many instances to the frontiers of knowledge. At the same time, we have endeavored to keep the mathematical prerequisities as low as possible, namely, knowledge of measure-theoretic probability and some acquaintance 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 in continuous time. We support this point of view by showing how by means of stochastic integration and random time change, all continuous martingales and a multitude of continuous 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 to be used as a tool when we later study passage times of Brownian motion.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jul 01, 1985
- Accession Number
- ADA160020
Entities
People
- I. Karatzas
- S. E. Shreve
Organizations
- Massachusetts Institute of Technology