I Probablity theory(金融随机分析-南京理工大学,陈萍)

2 , W,
HHH , HHT , HTH , HTT , THH , THT , TTH , TTT ,
AHH
AHT
ATH
ATT
and all sets which can be build by taking unions of these }
s AHH , AHT , ATH , ATT
( A B) ( A)( B), A , B
Definition 1.2.4: We say that two s-algebras G , H , are independent if
( A B) ( A)( B), A G , B H
ci X ( ) 0
Fi Fi
i 1 n
ci I Fi ( )
i 1
n
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Lemma 1.3.2 Every real valued measurable function X is the limit of a sequence {Xn}of simple functions; If f is non negative , then each Xn may be taken non negative and the sequence {Xn} may be assumed increasing.
uS S dS
d 2S
u 2 dS
Consider Sk1 ( B) : W Sk B
B B
udS
ud 2 S
k measurable?
Binomial tree
d 3S
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Probability measure induced by a random variable:
X 1 ( B); B (open interval on )
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• Random experiment of 3 coin tosses S k ( ) : stock price at time k
S0 ( ) S1 ( ) S 2 ( ) S ( ) 3
u3S
u2S
Sk : W
0 , W : trivity measure
Definition 1.1.2: A probability measure is a function mapping into [0, 1] with the following properties: 1. (W) 1 2. If A1 , A2 , is a sequence of disjoint sets in , then
3. Ak , Ak k 1
A pair (W, ) is called a measurable space. An element of is called a measurable subset of W • A s-algebra contains , so does W,
Stochastic calculus for finance
1 Probablity theory
Chen Ping
1
参考书： [1]Steven E Severve,Stochastic calculus for finance 2,Springer,2008 [2] Bernt ksendal ,Stochastic differential equations, An introduction with applications
Definition 1.2.5: We say that two random variables, X and Y, are independent if s-algabra generated by these random variables are independent, i.e., s(X) and s(Y) are independent.
n W
0 X1 ( ) X 2 ( ) X 3 ( )
4. General: X X X maxX ( ),0 max X ( ),0
1 Probablity theory
1.1 Probability and measure theory 1.2 Random variable,distribution 1.3. Integration using general probability measure 1.4 conditional Expectation 1.5 Stochastic process 1.6 martingales 1.7 Markov Processes
L X ( B) PX ( ) B --Distribution
In fact, the induced measure LX is a probability measure, because L X () 1
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Independence:
Definition 1.2.3: We say that two sets are independent if
n
The integral of X, is defined as
W
X ( )dP( ) c P( F )
i 1 i i
15
n
3. X is nonnegative:
Xd : sup lim
W
W
X n d X n ( ) X ( )
X n d

(Xn: simple function)
DEFINITION OF INTEGRAL i) A simple function X ( ) ci I F ( ) on a measure space i i 1 (W,F,P) is integrable if P(Fi)< , for every index i for which ci 0.
Ak ( Ak ) k 1 k 1
(W, , P) is called a probability triple;
a measurable subset of W is called an event.
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Example 1.1.3
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If two random variables, X and Y, are independent, then two functions, g(X) and h(Y), are also independent. Definition 1.2.6: Given (W, , {k }, P), a process X k is called adapted to the filtration {k } if for each k, X k is k measurable.
• For a random variable X on (W, , ), we write
X ( ) B W X ( ) B X 1 B ,
So
X ( ) B : X 1( B )
B
• The induced measure of B is a measure on ,Β s.t.
4
1.1 Probability and measure theory
Definition1.1.1: A s-algebra is a collection of subsets of W with the following three properties: 1. 1. If A , then its complement Ac
(t ) r (t ) " noise "
Where we do not know the exact behaviour of the noise term, only its probability distribution. the function r(t)is assumed to be nonrandom. How do we solve (1.1)in this case?
W HHH , HHT , HTH , HTT , THH , THT , TTH , TTT
1 , W,
HHH , HHT , HTH , HTT , THH , THT , TTH , TTT
H on the first toss AH T on the first toss AT
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1.3. Integration using general probability measure
1. Integral
Definition 1.3.1 A function X() , defined on a measurable space W is called simple if there is a finite , disjoint class {F1,...,Fn} of measurable sets and a finite set {c1,...,cn}of real numbers such that
Springer, 2000
2
Introduction
PROBLEM Consider the simple stock price model
dS t dt
t S t ,
S 0 x
(1.1)
Where S(t) is the stock price at time t, and μ(t) is the relative rate of growth at time t. It might happen that μ(t) is not completely known, but subject to some random environmental effects, so that we have
0 1 2
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• 0 contains no information • 1 contains the information up to time 1 (the first toss) • 2 contains the information up to time 2 (the first two tosses) Definition 1.1.4: A filtration is an increasing sequence of s-algebras w.r.t time s.t.
0 1 2 k
(W, , {k }, P) is called a filtered space, where 0 1
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1.2 Random variable,distribution
Definition 1.2.1: Given (W, , P), a function f : W is called measurable if
f 1 ( B) W f B
for all B ∈B ( )
A random variable X : W is an measurable function Definition 1.2.2: salgebra s(X) generated by a random variable X is the smallest s-algebra on W containing all the sets