金融随机分析全套课件184p

1. (W) 1 2. If A1, A2, is a sequence of disjoint sets in , then

Ak (Ak ) k1 k1
(W, , P) is called a probability triple; a measurable subset of W is called an event.
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Example 1.1.3
W HHH , HHT , HTH , HTT , THH , THT , TTH , TTT
1 , W, HHH , HHT , HTH , HTT , THH ,THT ,TTH ,TTT
Hale Waihona Puke H on the first toss AH

3. k1 Ak , Ak
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,
0 , W : trivial s-algebra
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Probability measure
Definition 1.1.2: A probability measure is a function mapping into [0, 1] with the following properties:
Stochastic calculus for finance
1 Probablity theory
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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 1.1.4: 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
s AHH , AHT , ATH , ATT
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)
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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
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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
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
T on the first toss AT
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 }
Springer, 2000
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Introduction
PROBLEM Consider the simple stock price model
dS t t S t , S 0 x (1.1)
dt
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
(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?