【笔记】Change of Measure
基于B站up Jerry Xu 的视频《概率测度变换》系列的笔记
Section 1:Preliminary of Probability Theory
1. Probability Space
(Ω,F,P)Sample Space (可能出现的结果):Ω
- Ω={ω1,ω2,…}
- Ω=ωi,i∈I(indexset),无限多不可数
- Random Variable X(ω)定义在Ω上
- X:Ω→RK
- X−1(B)={ω∈Ω:X(ω)∈B}∈Fforall
- Borel sets:B∈B(RK)
σ-algebra (all measurable events):F
- ∅∈F
- A∈F⇒AC∈F
- A1,A2,…∈F⇒∪∞n=1An∈F
- Events are subset of Ω
Probability measure (all events will be assigned a value):P
P(Ω)=1
IfA1,A2,…aredisjointsets
⇒P(∪∞n=1An)=∑∞n=1P(An)
P能给F中每一个事件都映射到闭区间0到1上,所以概率测度定义域是σ域
- P:Ω→[0,1]
De Morgan’s law
(A∩B)C=AC∪BC, (A∪B)C=AC∩BC2. Other Definition
期望
- Countable Infinite
- Uncountable Infinite
Riemann积分(upper)&Lebesgue积分(lower)
Indicator function (示性函数)
IA(ω)={1,ω∈A0,ω∉AE[IA(ω)]=P(ω∈A)积分
For A⊂Ω,A∈F
∫AX(ω)dP(ω)=∫ΩIA(ω)X(ω)dP(ω)For A⊂Ω,B⊂Ω,A,B∈F
and A∩B=∅
∫A∪BX(ω)dP(ω)=∫ΩIA∪B(ω)X(ω)dP(ω)=∫ΩIA(ω)X(ω)dP(ω)+∫ΩIB(ω)X(ω)dP(ω)(A∩B=∅) | IA | IB | IA∪B |
---|---|---|---|
A | 1 | 0 | 1 |
B | 0 | 1 | 1 |
A∪B | 0 | 0 | 0 |
3. Change of Measure
测度
测度P:真实世界测度
测度Q:单个风险中性测度 (股票过程) 多个等价鞅测度 (债券过程)
Radon-Nikodym Derivative
Let Z≥0, with
EP[Z(ω)]=1For A∈F, define
Q(A)=∫AZ(ω)dP(ω)Q(A)=∑ω∈AQ(ω)=∑ω∈AP(ω)Z(ω)Then Q is a prob measure.
Furtheremore, if X≥0, then
EQ[X]=EP[XZ]if Z>0, then
EQ[XZ]=EP[X]4. Example
Question:we have X(ω)=ω∼NP(0,1) . Set Y(ω)=X(ω)+θ(θ>0) . So it follow NP(θ,1) . Build a Probability measure Q and make Y follow NQ(0,1) . Find the Radon-Nikodym Derivative.
Z(ω)=dQ(ω)dP(ω)Recall that the PDF and CDF of Standard normal distribution:
CDF
Φ(x)=∫x−∞1√2πe−s22dsUnder P:X∼NP(0,1), we have:
ΔP(l)=P(l≤ω≤l+Δl)=Φ(l+Δl)−Φ(l)Under Q:Y∼NQ(0,1), we have:
ΔQ∗(l)=Q(l≤Y(ω)≤l+Δl)=Q(l−θ≤ω≤l+Δl−θ)=Φ(l+Δl)−Φ(l)Set l→l+θ
ΔQ(l)=ΔQ∗(l+θ)=Q(l≤ω≤l+Δl)=Φ(l+Δl+θ)−Φ(l+θ)Solving the function
Z(l)=dQ(l)dP(l)=limso
在随机变量中 Z(\omega) 为 exponential martingale (指数鞅)
真实世界中
风险中性世界中
Section 2:Girsanov’s Theorem
1. Conditional Expection
\mathscr{G} is a sub-\sigma-algebra of \mathcal{F}. X is a nonnegative & integrable r.v. The conditiond expextation of X given \mathscr{G}, denoted \mathbb{E}[X|\mathscr{G}], is any r.v. that satisfies:
- \mathbb{E}[X|\mathscr{G}] is \mathscr{G}-measurable
- \int_A \mathbb{E}[X|\mathscr{G}]d\mathbb{P}(\omega)=\int_AX(\omega)d\mathbb{P}(\omega), for \forall A\in\mathscr{G} : Partical Averaging
Properties
Pull out property: If X is a \mathscr{G}-measureable, then
Tower property (or Iteration): If \mathcal{H} is a sub-\sigma-algebra of \mathscr{G}
Radon-Nikodym Derivative Process
Z(t) is a martingale ( 0\leq s\leq t\leq T)
Z(t) is Radon-Nikodym Derivative (Y is \mathcal{F}(t)-measureable)
\mathbb{Q}^\mathbb{P}[Y|\mathcal{F}(s)]=\frac{1}{Z(s)}\mathbb{E}^\mathbb{P}[YZ(t)|\mathcal{F}(s)] (Right hand side is the conditiond expextation of Y)
- \frac{1}{Z(s)}\mathbb{E}^\mathbb{P}[YZ(t)|\mathcal{F}(s)] is \mathcal{F}(s)-measureable
- For any A \in \mathcal{F}(s), we have \int_AY(\omega)d\mathbb{Q}(\omega)=\int_A \frac{1}{Z(s)}\mathbb{E}^\mathbb{P}[YZ(t)|\mathcal{F}(s)]d\mathbb{Q}(\omega)
levy’s theorem:Let M(t), t\ge 0, be a martingale, relative to\mathcal{F}(t), t\ge 0. Assume M(0)=0, M(T) has contiunous paths and quadratic variation = t ( 二次变差 ) for all t\ge 0. \Rightarrow M(t) is a Brownian Motion.
当 \{t_i^n\}_{i=0}^n 遍取 [0,t] 的分割,其模 \delta_n=max_{0\leq i\leq n-1}\{t_{i+1}^n-t_i^n\}\to 0时,依概率收敛意义下的极限
BM 只有 dW(t)dW(t)=dt 会在一单位时间内积累一单位二次变差. dtdt=dtdW(t)=0
2. Girsanov
Let W(t) be a BM on (\Omega,\mathcal{F},\mathbb{P}) and let \mathcal{F}(t) be a filtration. For this BM, let \Theta(t) be an adopted process. Define
or
Set Z=Z(T). Then \mathbb{E}Z=1, Z\ge 0
Where \mathbb{Q} given by
The process \tilde{W}(t) is a BM
Prove
\mathbb{E}Z=1
Set
\Rightarrow Z(t) is a \mathbb{P}-martingale
\tilde{W}(t)Z(t) is a martingale
\Rightarrow \tilde{W}(t)Z(t) is a \mathbb{P}-martingale
\tilde{W}(t) is a martingale \mathbb{E}^\mathbb{Q}[\tilde{W}(t)|\mathcal{F(s)}]=\tilde{W}(s)
\Rightarrow \tilde{W}(t) is a \mathbb{Q}-martingale
\tilde{W}(0)=W(0)+\int_0^0\Theta(s)ds=0
The both parts of \tilde{W}(t) has contiunous paths.
d\tilde{W}(t)=dW(t)+\Theta(t)dt, where dtdt=0 accumulated zero quadratic variation.
[3-6]\Rightarrow \tilde{W}(t) is a BM