Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 14

Elements of Multidimensional Stochastic Analysis

 

 

In this book, we have essentially restricted ourselves to one-dimensional processes. For the reader’s convenience, in this chapter we present a short overview of the main definitions and facts in the multidimensional case. Some of them are rather simple and natural generalizations of the facts already stated, other facts can be checked without any principal difficulties, while for a deeper understanding of some facts, additional, more exhaustive sources will be necessary.

14.1. Multidimensional Brownian motion

A random process

with values in the k-dimensional real space , such that all coordinates Bⁱ, i = 1,2,… ,k, are independent one-dimensional Brownian motions is called a k-dimensional (standard) Brownian motion.

The set _t = _t^B of all random variables that depend only on the values of a k-dimensional Brownian B until moment t is called its history (or past) until moment t. If a random variable X ∈_t, then X B_u − B_s for ...