Optional Stopping Theorem(OST)

test

Proof. Define , then

is also a submartingale, and thus . This implies

To prove the second inequality, we define

is predictable and

This implies

Proof. It's enough to show . We can define a predictable process by

In addition, we have

Then implies

Proof. We can define a bounded stopping time by

since .

Since is a bounded stopping time, we have

This implise

Suppose is a submartingale such that , Then for any stopping times, is well defined.

Next, we suppose is u.i.

Proof. is still a submartingle with

Now the MCT(a.s) implies . For a fixed and we have

Since is u.i, we have . Let , we have

On the other hand, we have

This implise

Let , and using the fact that is u.i, we have

Then let , we have thus

Proof. Since are bounded stopping times, we have

Since is u.i., pass to , we have

and

Proof. For fixed , we have

In addition, we have

Since

Now pass to , we have