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Stochastic ProcessesDavid Nualart nualart@mat.ub.es

1

11.1

Stochastic ProcessesProbability Spaces and Random Variables

In this section we recall the basic vocabulary and results of probability theory. A probability space associated with a random experiment is a triple (, F, P ) where: (i) is the set of all possible outcomes of the random experiment, and it is called the sample space. (ii) F is a family of subsets of which has the structure of a -eld: a) F b) If A F , then its complement Ac also belongs to F c) A1 , A2 , . . . F = Ai F i=1 (iii) P is a function which associates a number P (A) to each set A F with the following properties: a) 0 P (A) 1, b) P () = 1 c) For any sequence A1 , A2 , . . . of disjoints sets in F (that is, Ai Aj = if i = j), P ( Ai ) = P (Ai ) i=1 i=1 The elements of the -eld F are called events and the mapping P is called a probability measure. In this way we have the following interpretation of this model: P (F )=probability that the event F occurs The set is called the empty event and it has probability zero. Indeed, the additivity property (iii,c) implies P () + P () + = P (). The set is also called the certain set and by property (iii,b) it has probability one. Usually, there will be other events A such that P (A) = 1. If a statement holds for all in a set A with P (A) = 1, then it is customary 2

to say that the statement is true almost surely, or that the statement holds for almost all . The axioms a), b) and c) lead to the following basic rules of the probability calculus: P (A B) = P (A) + P (B) if A B = P (Ac ) = 1 P (A) A B = P (A) P (B). Example 1 Consider the experiment of ipping a coin once. = {H, T } (the possible outcomes are Heads and Tails) F = P() (F contains all subsets of ) 1 P ({H}) = P ({T }) = 2 Example 2 Consider an experiment that consists of counting the number of trac accidents at a given intersection during a specied time interval. = {0, 1, 2, 3, . . .} F = P() (F contains all subsets of ) k P ({k}) = e (Poisson probability with parameter > 0) k! Given an arbitrary family U of subsets of , the smallest -eld containing U is, by denition, (U) = {G, G is a -eld, U G} . The -eld (U) is called the -eld generated by U. For instance, the -eld generated by the open subsets (or rectangles) of Rn is called the Borel -eld of Rn and it will be denoted by BRn . Example 3 Consider a nite partition P = {A1 , . . . , An } of . The -eld generated by P is formed by the unions Ai1 Aik where {i1 , . . . , ik } is an arbitrary subset of {1, . . . , n}. Thus, the -eld (P) has 2n elements. 3

Example 4 We pick a real number at random in the interval [0, 2]. = [0, 2], F is the Borel -eld of [0, 2]. The probability of an interval [a, b] [0, 2] is P ([a, b]) = ba . 2

Example 5 Let an experiment consist of measuring the lifetime of an electric bulb. The sample space is the set [0, ) of nonnegative real numbers. F is the Borel -eld of [0, ). The probability that the lifetime is larger than a xed value t 0 is P ([t, )) = et . A random variable is a mapping R X() which is F-measurable, that is, X 1 (B) F , for any Borel set B in R. The random variable X assigns a value X() to each outcome in . The measurability condition means that given two real numbers a b, the set of all outcomes for which a X() b is an event. We will denote this event by {a X b} for short, instead of { : a X() b}. A random variable denes a -eld {X 1 (B), B BR } F called the -eld generated by X. A random variable denes a probability measure on the Borel -eld BR by PX = P X 1 , that is, PX (B) = P (X 1 (B)) = P ({ : X() B}). The probability measure PX is called the law or the distribution of X. We will say that a random variable X has a probability density fX if fX (x) is a nonnegative function on R, measurable with respect to the Borel -eld and such thatb X

P {a < X < b} =a +

fX (x)dx,

for all a < b. Notice that fX (x)dx = 1. Random variables admitting a probability density are called absolutely continuous. 4

We say that a random variable X is discrete if it takes a nite or countable number of dierent values xk . Discrete random variables do not have densities and their law is characterized by the probability function: pk = P (X = xk ). Example 6 In the experiment of ipping a coin once, the random variable given by X(H) = 1, X(T ) = 1 represents the earning of a player who receives or loses an euro according as the outcome is heads or tails. This random variable is discrete with 1 P (X = 1) = P (X = 1) = . 2 Example 7 If A is an event in a probability space, the random variable 1A () = 1 if A 0 if A /

is called the indicator function of A. Its probability law is called the Bernoulli distribution with parameter p = P (A). Example 8 We say that a random variable X has the normal law N (m, 2 ) if b (xm)2 1 P (a < X < b) = e 22 dx 2 2 a for all a < b. Example 9 We say that a random variable X has the binomial law B(n, p) if n k P (X = k) = p (1 p)nk , k for k = 0, 1, . . . , n. The function FX : R [0, 1] dened by FX (x) = P (X x) = PX ((, x]) is called the distribution function of the random variable X. 5

The distribution function FX is non-decreasing, right continuous and withx

lim FX (x) = 0, lim FX (x) = 1.

x+

If the random variable X is absolutely continuous with density fX , then,x

FX (x) =

fX (y)dy,

and if, in addition, the density is continuous, then FX (x) = fX (x). The mathematical expectation ( or expected value) of a random variable X is dened as the integral of X with respect to the probability measure P : E(X) =

XdP .

In particular, if X is a discrete variable that takes the values 1 , 2 , . . . on the sets A1 , A2 , . . ., then its expectation will be E(X) = 1 P (A1 ) + 1 P (A1 ) + . Notice that E(1A ) = P (A), so the notion of expectation is an extension of the notion of probability. If X is a non-negative random variable it is possible to nd discrete random variables Xn , n = 1, 2, . . . such that X1 () X2 () andn

lim Xn () = X()

for all . Then E(X) = limn E(Xn ) +, and this limit exists because the sequence E(Xn ) is non-decreasing. If X is an arbitrary random variable, its expectation is dened by E(X) = E(X + ) E(X ), 6

where X + = max(X, 0), X = min(X, 0), provided that both E(X + ) and E(X ) are nite. Note that this is equivalent to say that E(|X|) < , and in this case we will say that X is integrable. A simple computational formula for the expectation of a non-negative random variable is as follows:

E(X) =0

P (X > t)dt.

In fact,

E(X) = =0

XdP = + 0

1{X>t} dt dP

P (X > t)dt.

The expectation of a random variable X can be computed by integrating the function x with respect to the probability law of X:

E(X) =

X()dP () =

xdPX (x).

More generally, if g : R R is a Borel measurable function and E(|g(X)|) < , then the expectation of g(X) can be computed by integrating the function g with respect to the probability law of X:

E(g(X)) =

g(X())dP () =

g(x)dPX (x).

The integral g(x)dPX (x) can be expressed in terms of the probability density or the probability function of X:

g(x)dPX (x) =

g(x)fX (x)dx, fX (x) is the density of X g(xk )P (X = xk ), X is discrete k

7

Example 10 If X is a random variable with normal law N (0, 2 ) and is a real number, E(exp (X)) = = = e 1 2 2 1 2 2 . e 2 2 2

ex e 22 dx

x2

e

(x 2 )2 2 2

dx

2 2 2

Example 11 If X is a random variable with Poisson distribution of parameter > 0, then

E(X) =n=0

n

e n e n1 = e = . n! (n 1)! n=1

The variance of a random variable X is dened by 2 = Var(X) = E((X E(X))2 ) = E(X 2 ) [E(X)]2 , X provided E(X 2 ) < . The variance of X measures the deviation of X from its expected value. For instance, if X is a random variable with normal law N (m, 2 ) we have P (m 1.96 X m + 1.96) = P (1.96 = (1.96) (1.96) = 0.95, where is the distribution function of the law N (0, 1). That is, the probability that the random variable X takes values in the interval [m1.96, m+ 1.96] is equal to 0.95. If X and Y are two random variables with E(X 2 ) < and E(Y 2 ) < , then its covariance is dened by cov(X, Y ) = E [(X E(X)) (Y E(Y ))] = E(XY ) E(X)E(Y ). A random variable X is said to have a nite moment of order p 1, provided E(|X|p ) < . In this case, the pth moment of X is dened by mp = E(X p ). 8 X m 1.96)

The set of random variables with nite pth moment is denoted by Lp (, F, P ). The characteristic function of a random variable X is dened by X (t) = E(eitX ). The moments of a random variable can be computed from the derivatives of the characteristic function at the origin: mn = for n = 1, 2, 3, . . .. We say that X = (X1 , . . . , Xn ) is an n-dimensional random vector if its components are random variables. This is equivalent to say that X is a random variable with values in Rn . The mathematical expectation of an n-dimensional random vector X is, by denition, the vector E(X) = (E(X1 ), . . . , E(Xn )) The covariance matrix of an n-dimensional random vector X is, by denition, the matrix X = (cov(Xi , Xj ))1i,jn . This matrix is clearly symmetric. Moreover, it is non-negative denite, that means,n

1 (n) (t)|t=0 , in X

X (i, j)ai aj 0i,j=1

for all real numbers a1 , . . . , an . Indeed,n n n

X (i, j)ai aj =i,j=1 i,j=1

ai aj cov(Xi , Xj ) = Var(i=1

ai Xi ) 0

As in the case of real-valued random variables we introduce the law or distribution of an n-dimensional random vector X as the probability measure dened on the