This is a sequel to Martingale Theory II: Conditional Expectation. Our aim here is to prove the main theorems about discrete time martingales. More advanced probability texts (e.g. those on stochastic calculus) assume that these theorems are well known to the reader.

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**6. Optional stopping theorem**

(Basically we follow [S].) Suppose you buy a stock at time 0, and the price process is a martingale. Suppose that you are going to sell the stock before next year, at a future date that you may decide (i.e. is a *random* time). Roughly speaking, (a version of) the **optional stopping theorem **says that if does not contain information about the future, then no matter how you choose , you will not earn money on the average, i.e. . Similar results, sometimes called the **gambling system theorems,** were proved to show the impossibility of devising a “winning strategy” for typical casino games. All these are initial motivations of martingale theory.

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How to formulate a “random time that does not contain information about the future”? First, we should have a **filtration** to represent our available information at each time. (In the above example, we may pick .) Then, for each time , the choice should be decidable using the information at time . In other words, we should have . This motivates the following

**6.1 Definition**. Let be measurable. is a **stopping time with respect to the filtration ** if for all , . (We also write: is an -stopping time. Sometimes we omit the filtration if it is understood.)

If you want to practice, do:

**6.2 Exercise**. Prove that is a -stopping time if and only if for all .

**6.3 Exercise.** Any (as a constant random variable) is a stopping time. If and are -stopping times, and , then are -stopping times. Find an example that is a stopping time, but is not.

**6.4 Example**. Recall the **gambler’s ruin problem** (see I), where and . Consider the **first hit** to or :

.

For each , we have the decomposition (check!)

.

This shows is (measurable and is) an -stopping time.

Now we prove that in Example 6.4, . That is, you will leave the casino in finite time *almost surely*. More importantly, this implies that is defined everywhere on except on a null set. The proof is beautiful *probabilistic thinking*: **if something is possible, and you try it again and again, it will certainly happen** (see Chapter 1 of [S]).

*Proof*: We partition the time set into

.

We write . Now, observe that

.

That is, the walk will certainly hit the boundary (and hence triggers ) if there are consecutive heads. We partition time in the above way because we now have:

The events are *independent*. (Check)

Also, . Hence, from

we may conclude from the second half of **Borel-Cantelli lemma** that with probability 1, occurs *infintely often*, i.e.

.

In particular, .

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To prove the optional stopping theorem, we introduce two auxillary concepts (which are also quite important themselves).

**6.5 Definition**. Let be a filtration on . A stochastic process is said to be **non-anticipating** with respect to if for all .

**6.6 Definition**. Let be a martingale, and be a stochastic process. The **martingale transform** of by is the stochastic process defined by

, .

To motivate the definition, suppose that is the stock price over time, and suppose you hold shares to stock between the time interval . The process represents your *strategy*. Clearly, should depend no extra information beyond . So is non-anticipating. Your accoumated profit/loss (P&L) is then the martingale transform of by . You may also think about it as a **discrete time stochastic integral** .

**6.7 Martingale transform theorem**. If is a martingale and is non-anticipating and each is bounded, then the martingale transform is also a martingale (all with respect to a filtration ).

*Proof*: Since is non-anticipating, it is easy to see that is adapted to . Also, since the s are bounded, each is integrable. Hence, it remains to verify that . We have

.

Since , we may pull it out and get

.

This shows that is a martingale.

**6.8 Remark**. There are two ways to interpret the martingale transform theorem. First, it is a **gambling system theorem**, stating that you cannot hope to profit in a “fair game”. (Think about as the stock price, and your strategy.) Second, it says that the class of martingale is *closed* under martingale transforms (or stochastic integration) by appropriate processes. This property motivates generalization in continuous time.

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For a martingale , we always have . The optional stopping theorem says that we may replace by a bounded stopping time. We will prove it using the martingale transform theorem. The idea is the following. Given a stopping time , we may choose a non-anticipating strategy that says: sell at time .

**6.9 Optional stopping theorem**. Let be a martingale and be a *bounded* stopping time, with respect to a filtration . Then .

We recall that, in general, is defined by on the (measurable) set .

(There are optional stopping theorems for sub/supermartingales, which we skip here.)

*Proof*: Let be a stopping time bounded by , i.e. for all . For each time , we let

if , and if .

In other words, . Note that, since , is (bounded and) non-anticipating.

Next, we observe that the (*shifted*) martingale transform

equals the **stopped process **. For, on the set we have

,

while on the set we have

.

The **martingale transform theorem **then implies that the process is a martingale (verify that the shift does not matter). So for all . Now since , we have . Finally we put to get

.

The theorem is proved.

**6.10 Remark**. To see that boundedness of is necessary, consider the simple random walk starting at and . It can be shown that is a stopping time and is finite with probability . Then and .

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**Solution of the gambler’s ruin problem**

Now we can solve the gambler’s ruin problem (see Section 1) rigorously. Recall that we would like to find , where . We have shown above that , hence is well-defined on a set with probability 1.

At first sight, the optional stopping theorem is not applicable because is obviously not bounded above ( can go up and down, up and down, etc…). The following trick (or technique) is frequently used:

To deal with this, let be fixed. Then is a *bounded* stopping time (see Exercise 6.3). Now we may apply the **optional stopping theorem **and get

.

Now, since , almost surely we have . Hence almost surely. Moreoveer, for all . Hence by the classical **bounded convergence theorem **we obtain

.

On the other hand we have, by definition of ,

.

Combining, we have .

**6.11 Exercise**. Consider the martingale and show that . What else can be calculated by martingale method?

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In the next post we will treat the martingale convergence theorems.