用户: Solution/ 笔记: 金融中的随机分析

此处参考由 Steven E. Shreve 所编写的英文教材《Stochastic Calculus for Finance II》, 此教材也有中译, 但作者用的是英文版, 故此处仅对名词做翻译.

1一些定义

$\sigma$-algebra $\sigma$-代数

定义 1.1. We say that $\mathcal{F}$ is a $\sigma$-algebra if $\mathcal{F}$ is a nonempty collection of subsets of $\Omega$ that satisfy:

1.	if $A\in \mathcal{F}$, then $A^{\complement }\in \mathcal{F}$ ;
2.	if $\forall i\in {\mathcal{I}}_{\text{countable}}$, $A_i\in \mathcal{F}$, then ${\cup }_{i\in \mathcal{I}}{A}_i\in \mathcal{F}$.

Filtration 域流

定义 1.2. Given a fixed number $T>0$, if :

•	$\forall 0\le t\le T$, $\mathcal{F}(t)$ is $\sigma$ - algebra ;
•	$\forall 0\le s\le t\le T$, $\mathcal{F}(s)\subset \mathcal{F}(t)$

Then,$$\mathbb{F}=\{\mathcal{F}(t)|t\in [0,T]\}$$is a filtration.

Martingale 鞅

定义 1.3. Given a fixed number $T>0$, filtration $\mathcal{F}(t)\in \mathbb{F}$ and a stochastic process $M(t)$, if $\forall 0\le s\le t\le T$: $$\mathbb{E}[M(t)|\mathcal{F}(s)]=M(s)$$Then $M(t)$ is a martingale.

Adapted stochastic process 适应随机过程

定义 1.4. We say that stochastic process $\Delta (t)$ is adapted to $\mathcal{F}(t)$ if $\Delta (t)$ is $\mathcal{F}(t)$-measurable.

Markov process 马尔科夫过程

定义 1.5. Given a fixed number $T>0$, filtration $\mathcal{F}(t)\in \mathbb{F}$ , an adapted stochastic process $X(t)$ and Borel-measurable functions $f,g$ , if $\forall 0\le s\le t\le T$: $$\mathbb{E}[f\left(X(t)\right)|\mathcal{F}(s)]=g\left(X(s)\right)$$Then $X(t)$ is a Markov process.

2Symmetric Random Walk 对称随机游走

We consider a coin flip game with a 50-50 probability of heads or tails. Let $j$-th times game be $X_j$, thus we have:$$X_j=\{\begin{array}{l}1,\text{if}{w}_j=H\\ -1,\text{if}{w}_j=T\end{array}$$It is easy to prove that:$$\begin{array}{rl}\mathbb{E}(X_j)& =\frac{1}{2}\times 1+\frac{1}{2}\times (-1)=0\\ \text{Var}(X_j)& =\mathbb{E}(X_j^2)=\frac{1}{2}\times 1^2+\frac{1}{2}\times (-1{)}^2=1\end{array}$$Then, we can define symmetric random walk:

定义 2.1. $M_k$ is said to be a symmetric random walk, that is, if: $$M_k=\begin{array}{r}\{\begin{array}{ll}{\sum }_{j=1}^k{X}_j& ,k\ge 1\\ 0& ,k=0\end{array}\end{array}$$

Its expected value and variance can be calculated by the work we done before:$$\begin{array}{rl}\mathbb{E}(M_k)& =\sum _{j=1}^k\mathbb{E}(X_j)=0\\ \text{Var}(M_k)& =\sum _{j=1}^k\text{Var}(X_j)=k\end{array}$$We can prove symmetric random walk is a martingale:$$\begin{array}{rl}\mathbb{E}(M_k|{\mathcal{F}}_s)& =\mathbb{E}(M_k-M_s|{\mathcal{F}}_s)+\mathbb{E}(M_s|{\mathcal{F}}_s)\\ & =\mathbb{E}(M_k-M_s)+M_s\\ & =M_s\end{array}$$Next, we define a quadratic variation:$$[M,M{]}_k=\sum _{j=1}^k(M_j-M_{j-1}{)}^2$$For symmetric random walk, its quadratic variation equals to $k$.

Besides, we can define Scaled Symmetric Random Walk:

定义 2.2. For fixed $n\in \mathbb{N}$, define: $$W^{(n)}(t)=\frac{}{\sqrt{n}}{M}_{nt}=\frac{1}{\sqrt{n}}\sum _{j=1}^{nt}{X}_j$$where $nt$ is an integer.

If $nt$ is not an integer, we consider the