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Introduction• We used a positive random variable Z to change probability measures on a
space Ω.
• is risk-neutral probability measure• P(ω) is the actual probability measure
• When Ω is uncountably infinite and then it will division by zero. We could rewrite this equation as
• But the equation tells us nothing about the relationship among and Z. Because , the value of Z(ω)
• However, we should do this set-by-set, rather than ω-by-ω
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Theorem 1.6.1• Let (Ω, F, P) be a probability space and let Z be an almost surely
nonnegative random variable with EZ=1. For A F define
Then is a probability measure. Furthermore, if X is a nonnegative random variable, then
If Z is almost surely strictly positive, we also have
for every nonnegative random variable Y.
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Proof of Theorem 1.6.1 (1)• According to Definition 1.1.2, to check that is a probability measure, we
must verify that and that is countably additive. We have by assumption
• For countable additivity, let A1, A2,… be a sequence of disjoint sets in F, and define , . Because
and , we may use the Monotone Convergence Theorem, Theorem 1.4.5, to write
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Proof of Theorem 1.6.1 (2)• But , and so
• Now support X is a nonnegative random variable. If X is an indicator function X=IA , then
which is
• When Z>0 almost surely, is defined and we may replace X in
by to obtain
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Definition 1.6.3• (測度等價 )• Let Ω be a nonempty set and F a σ-algebra of subsets of Ω. Two
probability measures P and on (Ω, F) are said to be equivalent if they agree which sets in F have probability zero.
•
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Definition 1.6.3 - Description• Under the assumptions of Theorem 1.6.1, including the assumption that
Z>0 almost surely, P and are equivalent. Support is given and P(A)=0. Then the random variable IAZ is P almost surely zero, which implies
On the other hand, suppose satisfies . Then almost surely under , so
Equation (1.6.5) implies P(B)=EIB=0. This shows that P and agree which sets have probability zero.
•
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Example 1.6.4 (1)• Let Ω=[0,1], P is the uniform (i.e., Lebesgue) measure, and
Use the fact that dP(ω)=dω, then
B[0,1] is a σ-algebra generated by the close intervals.Since [a,b] [0,1] implies
This is with Z(ω)=2ω
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Example 1.6.4 (2)• Note that Z(ω)=2ω is strictly positive almost surely (P{0}=0), and
According to Theorem 1.6.1, for every nonnegative random variable X(ω), we have the equation
This suggests the notation
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Example 1.6.4 - Description• In general, when P, , and Z are related as in Theorem 1.6.1, we may rewrite
the two equations and as
A good way to remember these equations is to formally write Equation is a special case of this notation that captures the idea behind the nonsensical equation that Z is somehow a “ratio of probabilities.”
• In Example 1.6.4, Z(ω) is in fact a ration of densities:
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Definition 1.6.5• (Radon-Nikodým derivative)• Let (Ω, F, P) be a probability space, let be another probability measure
on (Ω, F) that is equivalent to P, and let Z be an almost surely positive random variable that relates P and via (1.6.3). Then Z is called the Radon-Nikodým derivative of with respect to P, and we write
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Example 1.6.6 (1) • (Change of measure for a normal random variable)• X~N(0,1) with respect to P, Y=X+θ~N(θ,1) with respect to P.
Find such that Y~N(0,1) with respect to
SolFind Z>0, EZ=1, Let , and Z>0 is obvious because Z is defined as an exponential.
• And EZ=1 follows from the integration
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Example 1.6.6 (2)• Where we have made the change of dummy variable y=x+θ in the last
step.
• But , being the integral of the standard normal density, is equal to one.
where y=x+θ.• It shows that Y is a standard normal random variable under the
probability .• 當機率分配函數定義下來之後,一切特性都定義下來了。
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