【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,iI(indexset),无限多不可数
    • Random Variable X(ω)定义在Ω
      • XΩRK
      • X1(B)={ωΩ:X(ω)B}Fforall
        • Borel sets:BB(RK)
  • σ-algebra (all measurable events):F

    • F
    • AFACF
    • A1,A2,Fn=1AnF
    • 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

      • PΩ[0,1]

De Morgan’s law

(AB)C=ACBC, (AB)C=ACBC

2. Other Definition

期望

  • Countable Infinite
EX=k=1X(ωk)P(ωk)
  • Uncountable Infinite
EX=ΩX(ω)dP(ω)(Leb Integral)=xf(x)dx=xdF(x) pdf cdf

Riemann积分(upper)&Lebesgue积分(lower)

Indicator function (示性函数)

IA(ω)={1,ωA0,ωAE[IA(ω)]=P(ωA)

积分

For AΩ,AF

AX(ω)dP(ω)=ΩIA(ω)X(ω)dP(ω)

For AΩ,BΩ,A,BF

and AB=

ABX(ω)dP(ω)=ΩIAB(ω)X(ω)dP(ω)=ΩIA(ω)X(ω)dP(ω)+ΩIB(ω)X(ω)dP(ω)
(AB=) IA IB IAB
A 1 0 1
B 0 1 1
AB 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 Z0, with

EP[Z(ω)]=1

For AF, define

Q(A)=AZ(ω)dP(ω)Q(A)=ωAQ(ω)=ωAP(ω)Z(ω)

Then Q is a prob measure.

Furtheremore, if X0, then

EQ[X]=EP[XZ]

if Z>0, then

EQ[XZ]=EP[X]

4. Example

Question:we have X(ω)=ωNP(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:

PDF

ϕ(x)=12πex22, xR

CDF

Φ(x)=x12πes22ds

Under P:XNP(0,1), we have:

ΔP(l)=P(lωl+Δl)=Φ(l+Δl)Φ(l)

Under Q:YNQ(0,1), we have:

ΔQ(l)=Q(lY(ω)l+Δl)=Q(lθωl+Δlθ)=Φ(l+Δl)Φ(l)

Set ll+θ

ΔQ(l)=ΔQ(l+θ)=Q(lωl+Δl)=Φ(l+Δl+θ)Φ(l+θ)

Solving the function

Z(l)=dQ(l)dP(l)=lim

so

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

  1. Pull out property: If X is a \mathscr{G}-measureable, then

  2. Tower property (or Iteration): If \mathcal{H} is a sub-\sigma-algebra of \mathscr{G}

Radon-Nikodym Derivative Process

  1. Z(t) is a martingale ( 0\leq s\leq t\leq T)

  2. Z(t) is Radon-Nikodym Derivative (Y is \mathcal{F}(t)-measureable)

  3. \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)
  4. 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.

    • \{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

  1. \mathbb{E}Z=1

    Set

    \Rightarrow Z(t) is a \mathbb{P}-martingale

  2. \tilde{W}(t)Z(t) is a martingale

    \Rightarrow \tilde{W}(t)Z(t) is a \mathbb{P}-martingale

  3. \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

  4. \tilde{W}(0)=W(0)+\int_0^0\Theta(s)ds=0

  5. The both parts of \tilde{W}(t) has contiunous paths.

  6. d\tilde{W}(t)=dW(t)+\Theta(t)dt, where dtdt=0 accumulated zero quadratic variation.

  7. [3-6]\Rightarrow \tilde{W}(t) is a BM


Reference

  1. Jerry Xu B站主页
  2. 从Riemann积分到Lebesgue积分

【笔记】Change of Measure
http://achlier.github.io/2021/05/23/Change_of_Measure/
Author
Hailey
Posted on
May 23, 2021
Licensed under