Jean-François Le Gall - Brownian Motion, Martingales, and Stochastic Calculus

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Springer International Publishing Switzerland 2016

Jean-Franois Le Gall Brownian Motion, Martingales, and Stochastic Calculus Graduate Texts in Mathematics 10.1007/978-3-319-31089-3_1

1. Gaussian Variables and Gaussian Processes

Jean-Franois Le Gall 1

(1)

Dpartement de Mathmatiques, Universit Paris-Sud, Orsay Cedex, France

Gaussian random processes play an important role both in theoretical probability and in various applied models. We start by recalling basic facts about Gaussian random variables and Gaussian vectors. We then discuss Gaussian spaces and Gaussian processes, and we establish the fundamental properties concerning independence and conditioning in the Gaussian setting. We finally introduce the notion of a Gaussian white noise, which will be used to give a simple construction of Brownian motion in the next chapter.

1.1 Gaussian Random Variables

Throughout this chapter, we deal with random variables defined on a probability space . For some of the existence statements that follow, this probability space should be chosen in an appropriate way. For every real p 1, , or simply L p if there is no ambiguity, denotes the space of all real random variables X such that| X | p is integrable, with the usual convention that two random variables that are a.s. equal are identified. The space L p is equipped with the usual norm.

A real random variable X is said to be a standard Gaussian (or normal ) variable if its law has density

with respect to Lebesgue measure on . The complex Laplace transform of X is then given by

To get this formula (and also to verify that the complex Laplace transform is well defined), consider first the case when :

This calculation ensures that E [e zX ] is well-defined for every , and defines a holomorphic function on . By analytic continuation, the identity , which is true for every , must also be true for every .

By taking , , we get the characteristic function of X :

From the expansion

as (this expansion holds for every n 1 when X belongs to all spaces L p , , which is the case here), we get

and more generally, for every integer n 0,

If and , we say that a real random variable Y is Gaussian with -distribution if Y satisfies any of the three equivalent properties:

(i)

, where X is a standard Gaussian variable (i.e. X follows the -distribution);

(ii)

the law of Y has density

(iii)

the characteristic function of Y is

We have then

By extension, we say that Y is Gaussian with -distribution if Y = m a.s. (property (iii) still holds in that case).

Sums of independent Gaussian variables

Suppose that Y follows the -distribution, Y follows the -distribution, and Y and Y are independent. Then Y + Y follows the -distribution. This is an immediate consequence of (iii).

Proposition 1.1

Let (X n ) n1 be a sequence of real random variables such that, for every n 1, X n follows the

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