Change of Probability Measure by Radon-Nicodym Derivative and Some Applications

Jerry-Xu

---

（I）Preliminary of Probability Theory

i. Probability Space

柯尔莫哥洛夫三元组（Kolmogorov Triple）

$$\left( \Omega, \mathcal{F}, \mathbb{P} \right)$$

（a）Sample Space 样本空间

有限集（可测集）：

$$\Omega = \{w_{1}, w_{2}, w_{3}, ..., w_{n}\}$$

无限集（不可测集）：

$$\Omega = \{ w_{1}, w_{2}, w_{3}, ..., \}$$

$$\Omega = \{ w_{i}|i \in I \}, I := index\ set \left( 指标集 \right)$$

Example

Sample Space include:

$$HH:w_{1},HT:w_{2},TH:w_{3},TT:w_{4}$$

（b）$\sigma$-algebra $\sigma$-代数

Collection $\mathcal{F}$ is a $\sigma$-algebra which inclueds all measurable events.

event := 样本点的集合，它为样本点的某一组合，event 是 Sample Space 的子集（Subset）

Example

$$Event_{1}:No\ H = \{ w_{4} \}$$

$$Event_{2}:Only\ one\ H = \{ w_{2}, w_{3} \}$$

需满足以下三条条件

条件 $1$

包含空集（不可能事件）

$$\varnothing \in \mathcal{F}$$

条件 $2$

补运算得到的事件是有概率的

$$A \in \mathcal{F},\ A^{C} \in \mathcal{F}$$

条件 $3$

交、并运算得到的事件是有概率的

$$\text{If } A_{1}, A_{2}, ... \in \mathcal{F},\ \bigcup_{n=1}^{\infty} A_{n} \in \mathcal{F}$$

#

（c）Probability Measure 概率测度 $\mathbb{P}$

概率测度 $\mathbb{P}$ 本质上是一种映射（函数），它将样本空间映射到 $[0,1]$ 闭区间上。

All events will be assigned a value in $\left[ 0, 1 \right]$

$$\Omega \rightarrow \left[ 0, 1 \right]$$

Example

When

$$H: \frac{1}{3},\ T: \frac{2}{3}$$

We have

$$w_{1}:HH\ \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$$

$$w_{1} \stackrel{\mathbb{P}}\longrightarrow \frac{1}{9} \equiv \mathbb{P}\left( w_{1} \right) = \frac{1}{9} \in \left[ 0, 1 \right]$$

$$w_{2}:HT\ w_{2} \frac{1}{3} \times \frac{2}{3}=\frac{1}{9}$$

$$w_{2} \stackrel{\mathbb{P}}\longrightarrow \frac{2}{9} \equiv \mathbb{P}\left( w_{1} \right) = \frac{2}{9} \in \left[ 0, 1 \right]$$

且需要满足以下二条条件

条件 ($1$)

$$\mathbb{P} \left( \Omega \right) = 1\ or\ \sum_{w \in \Omega}\mathbb{P} \left( w \right) = 1$$

条件 ($2$) - 可数可加性（Contable Additivity）

If $A_{1}$, $A_{2}$, ..., $A_{n}$, ... are disjoint sets, then we have

$$\mathbb{P} \left( \bigcup_{n=1}^{\infty} A_{n} \right) = \sum_{n=1}^{\infty}\mathbb{P} \left( A_{n} \right)$$

适用于有限集或者无限集

条件 ($3$)

#

推论 $1$

$$\mathbb{P} \left( \varnothing \right) = 0$$

证明

$$\mathbb{P} \left( \varnothing \bigcup \Omega \right) = \mathbb{P} \left( \varnothing \right) + \mathbb{P} \left( \Omega \right)$$

$$\mathbb{P} \left( \Omega \right) = \mathbb{P} \left( \varnothing \right) + \mathbb{P} \left( \Omega \right)$$

$$\mathbb{P} \left( \varnothing \right) = 0$$

推论 $2$

$$\mathbb{P} \left( \bigcup_{n=1}^{N} A_{n} \right) = \sum_{n=1}^{N}\mathbb{P} \left( A_{n} \right)$$

证明

我们取

$A_{1}, A_{2}, ..., A_{n}, A_{n+1} = \varnothing, A_{n+2} = \varnothing, ...$

则有

$$\mathbb{P} \left( A_{1} \bigcup A_{2} \bigcup ... \bigcup A_{n} \bigcup A_{n+1} \bigcup ... \right) = \sum_{n=1}^{N}\mathbb{P} \left( A_{n} \right) + \mathbb{P} \left( \varnothing \right) + ...$$

$$\mathbb{P} \left( A_{1} \bigcup A_{2} \bigcup ... \bigcup A_{n} \right) = \mathbb{P} \left( \bigcup_{n=1}^{N} A_{n} \right) = \sum_{n=1}^{N}\mathbb{P} \left( A_{n} \right)$$

推论 $3$

（II）Rado-Nicodym Derivate and Examples

（III）A Critical Application

i. Girsanov's Theorm

ii. Market Price of Risk

iii. Risk Neutral Probability Measure

vi. Black-Shole Model

（IV）Feynman-Kac's Theorm

i. Transforming PDF into SDE or SDE into PDE

ii. Basic Monto Carlo Computing

iii. Examples of Monto Carlo Simulation

iv.

#