Martingale Convergence Theorem (MCT)

Given a stochastic process , e can define recursively by

Define a predictable process by

如图

我们可以发现代表buy low and sell high的收益, 定义

代表截止时刻, 上穿发生的次数. 每次上穿的收益肯定大于(b-a), 下界无法控制, 所以我们考虑过程, 容易证明

这就得到了上穿不等式.

upcrossing theorem

If is a submartingale, then

Proof. See above.

MCT, a.s.

MCT, a.s

If is a submartingale with , then converges a.s. to .

Proof. Define , The upcrossing theorem implies

Since the last expression holds for all rationales , we have

Thus exists and we denote it by . Now there remains to show .

On one hand, we have by Fatou's lemma. On the other hand,

Theorem

Suppose is a martingale with (p>1), then

Proof. Since

This implies and thus . Note that

Then DCT implies . In addition

It follows from the DCT that