# 金融工程自学笔记（2）：伊藤积分(金融工程自考本科)

## 2 Ito Calculus

（1） Riemann积分在 收敛，而伊藤积分在 收敛。

（2） 在Riemann积分中咱们一般这样迫临积分值：设

,

,

,

.

,

.

## 2.1 Definition

We shall call a random step process if there is a finite sequence of numbers and square integrable random variables such that

,

where is -measurable for . The set of random step processes will be denoted by .

The stochastic integral of a random step process is defined by:

For any random step processes ,

for any and any ,

.

## 2.2 General Case

We denote by the class of stochastic process such that

,

and there is a sequence of random step processes such that

.

We call the Ito stochastic integral (from 0 to ) of if

For any the stochastic integral exists, is unique (as an element of , i.e. to within equality a.s.) and satisfies

## 2.3 Examples

Problem 1:

，且 ，即 平分这一段区间。咱们令

Problem 2:

Problem 3: