以及

定义

证明

进而证明，当分割的网格趋于零时，

在  中成立。

最后证明，布朗运动几乎所有样本路径在  上都具有无限总变差。

解答

因为

所以

根据正态随机变量的四阶矩公式，

因此

另一方面，

由于不同时间区间上的布朗增量相互独立，所以随机变量  也相互独立。因此

记分割的网格为

则

所以

当  时，

因此

这说明布朗运动的二次变差为

下面证明总变差为无穷。

假设某条连续函数  在  上具有有限总变差

对于任意分割，

由于  在紧区间上连续，所以它一致连续。当分割网格趋于零时，

因此有限变差连续函数的二次变差必须为零。

但是布朗运动的二次变差为

所以布朗运动的样本路径不可能具有有限总变差。

因此，对于任意 ，

English Version

Chapter 2 Exercises and Detailed Solutions

Preliminary notation

Let  be a probability space. Let  denote the expectation of a random variable .

A one-dimensional standard Brownian motion satisfies

and

Its increments over disjoint time intervals are independent.

Exercise 2.1: Countably valued random variables

Problem

Suppose that  takes only countably many values

Prove that  is a random variable if and only if

for every . Derive formulas for , , and .

Solution

If  is measurable, then each singleton  is a Borel set. Hence

Conversely, suppose that every set  is measurable. For any Borel set ,

The right-hand side is a countable union of measurable sets, so  is measurable.

Pointwise,

By the monotone convergence theorem,

If , then

For a bounded measurable function ,

and therefore

Exercise 2.2: Distribution functions and the density of 

Solution

The distribution function

satisfies

Since the events  increase to ,

Since the events  decrease to the empty set,

If $x_1<x_2$, then<="" p="">

so  is nondecreasing.

For ,

Therefore,

which proves right continuity.

If  is measurable and , then

Now set

Since ,

For ,

Differentiating gives

By symmetry,

Therefore,

Exercise 2.3: Intersections of sigma-algebras

Let

Every  contains , so .

If , then  for every , and hence

for every . Thus .

If , then

for every . Hence

Therefore  is a sigma-algebra.

Exercise 2.4: Chebyshev's inequality

On the event ,

Hence

Taking expectations gives

Thus

Similarly, on ,

Therefore,

and

Exercise 2.5: Products of independent random variables

For simple independent random variables,

independence gives

Thus

General bounded random variables can be uniformly approximated by simple random variables. Passing to the limit gives

Exercise 2.6: The Borel–Cantelli lemma

Define

Then  decreases and

Since the series converges, its tail tends to zero. Hence

By continuity from above,

Thus infinitely many  occur with probability zero.

Exercise 2.7: Finite sigma-algebras

Suppose  form a partition of . The family of all unions of the  contains , is closed under complements, and is closed under countable unions. Hence it is a sigma-algebra.

Conversely, every finite sigma-algebra can be decomposed into finitely many atoms

Every measurable set is a union of these atoms.

If  is measurable, then it must be constant on every atom. Therefore,

Hence  takes only finitely many values.

Exercise 2.8: Moments of Brownian motion

Since ,

Expanding both sides and comparing coefficients gives

and

In particular,

For an -dimensional Brownian motion, let

where the  are independent  variables. Then

Exercise 2.9: Versions and continuous paths

For every fixed ,

Hence  almost surely.

For finitely many times ,

Thus  and  have the same finite-dimensional distributions.

However, for each fixed ,

is discontinuous at , whereas

is continuous everywhere.

Exercise 2.10: Stationary increments

Define

For