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》, 也有中译版, 这边仅对名词做中译

1一些定义

Filtration 域流

定义 1.1. 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 1.2. 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, also known as an adapted stochastic process.

Markov process 马尔科夫过程

定义 1.3. 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