复旦金融随机过程课件

P(A).
∞ F , we have: P(fAn gn =1 ) =
∞ 3. For disjoint events fAn gn =1
n =1
∑ P ( An ) .
∞
C. Ma (Institute)
MP
09/10
6 / 22
Introduction
A random variable de…ned on a probability space (Ω, F , P) is a function x : Ω ! X taking values in a second measure space (X, B(X)). X is an Euclidean space, where B(X) is the smallest σ-algebra that contains all open sets of X.
Lecture 1: Mathematics Preliminaries
Chenghu Ma
School of Management, Fudan University
ptember 2011
C. Ma (Institute)
MP
09/10
1 / 22
Contents
1
Contents State space Random Variable CDF and PDF Expectation and conditional expectation Independence Characteristic Functions Norm Space Convergence
3 4
C. Ma (Institute)
MP
09/10
3 / 22
State Space
The uncertainty is about the realization of state of nature in a state space Ω. Ω is the state space with its subsets called events. The time-t information consists of all events observable at time-t. The information …ltration F = fFt gt 2T is a σ-algebra on Ω satis…es
C. Ma (Institute)
MP
09/10
7 / 22
CDF and PDF
De…nition 1
CDF: the cumulative distribution of x is de…ned as: F (x ) = P(fω : x (ω ) < x g), for all x 2 X .
s |x
f (x )dx, s 2 Z .
The characteristic function is, in general, not de…ned on the entire complex plane. The domain of Θ(s ) is a strip region on Z m . We can use the Laplace inversion formula to deduce the p.d.f. function f . For one dimensional case, we have: 1 f (x ) = 2πj
Properties of expectation and conditional expectation:
1 2
For all x, jEP [x ]j
For all concave function f : X ! R, we have, Jensen’ inequality: s EP [f (x )] f (EP [x ]).
C. Ma (Institute)
MP
09/10
11 / 22
Expectation and conditional expectation (cont.)
For any random variable x, the expectation is de…ned by EP [x ] = EP [x + ] EP [x ]
The conditional expectation of x conditional on event A 2 F is de…ned by EP [x j A] = EP(.jA ) [x ]
C. Ma (Institute)
MP
09/10
12 / 22
Expectation and conditional expectation (cont.)
Properties
1. For all t, Ω 2 Ft . 2. A 2 Ft implies Ac = fω 2 Ω : ω 2 Ag. / ∞ ∞ 3. fAn gn =1 Ft implies [n =1 An 2 Ft .
C. Ma (Institute)
MP
09/10
4 / 22
State Space (cont.)
i =1
∑ pi δ ( x
n
xi ),
where pi is the probability x takes value xi . δ(x ) is Dirac-function that
Z
X
g (x ) δ (x
y ) dx = g (y ),
where g is continuous function on X, X 2 B(X).
C. Ma (Institute) MP 09/10 14 / 22
Characteristic Functions
For any given Rm -valued random variable x, its characteristic function Θ : Z m ! Z is de…ned as the Laplace transform of its density function f( ) Z Θ(s ) = Lff (x )g(s ) e
n 2
exp
1 (x 2
µ )T Σ
1
(x
µ) ,
n positive de…nite matrix.
C. Ma (Institute)
MP
09/10
10 / 22
Expectation and conditional expectation
For simple random variable x, the expectation of x is de…ned by EP [ x ] =
The information …ltration fFt gt 2T as a σ-algebra satis…es
Properties
1. For all ω 2 Ω, ω 2 Ft (ω ). 0 0 2. Ft is a partition of Ω: for all ω and ω , either Ft (ω ) \ Ft (ω ) = ∅ or 0 Ft ( ω ) = Ft ( ω ). 3. The information …ltration F = fFt gt 2T is monotonically increasing.
C. Ma (Institute)
MP
09/10
5 / 22
State Space (cont.)
Probability measure is a function P : F ! [0, 1] satis…es
Properties 1. P(Ω) = 1. 2. For all A 2 F , P(Ac ) = 1
De…nition 2
The density function f (.), if exists, can be expressed as: P(x 2 X ) =
Z
f (x )dx.
X
C. Ma (Institute)
MP
09/10
8 / 22
Examples
For random variable x with support at x1 , x2 , . . . , xn , the distribution function F ( ) is a step function with discontinuities at x1 , x2 , . . . , xn .The density function is f (x ) =
C. Ma (Institute)
MP
09/10
9 / 22
Examples
A n-dimensional random variable x has density function: f (x ) = 1
1 2
where µ is n-dimensional vector, Σ is n
(2π ) jΣj
EP [jx j] .
3
Let G and H be sub-σ-algebra of F . If G
H, then we have
EP [EP [x j H] j G] = EP [x j G].
4
The conditional expectation EP [x j y ] is F y measurable, and can be expressed as a B(X)-measurable function of y such that EP [ x j y ] = f ( y ) .
Properties 1. EP [x j y ] = EP [x ]. 2. EP [xy ] = EP [x ]EP [y ]; or, equivalently, Cov (x, y ) = 0. 3. Let f (x ), g (y ) and h (x, y ) be respectively the density functions of x, y , and the vector (x, y ). x and y are independent if and only if h (x, y ) = f (x )g (y ).