The next four lectures will be devoted to the foundational theorems of the theory of continuous time martingales. All of these theorems are due to Joseph Doob.

The following first theorem shows that martingales behave in a very nice way with respect to stopping times.

**Theorem (Doob’s stopping theorem)** *Let be a filtration defined on a probability space and let be a stochastic process that is adapted to the filtration , whose paths are right continuous and locally bounded. The following properties are equivalent:*

- is a martingale with respect to the filtration ;
- For any, almost surely bounded stopping time of the filtration such that , we have .

**Proof:**

Let us assume that is a martingale with respect to the filtration , whose paths are right continuous and locally bounded. Let now be a stopping time of the filtration that is almost surely bounded by . Let us first assume that takes its values in a finite set: . Thanks to the martingale property, we have

.

The theorem is therefore proved if takes its values in a finite set. If takes an infinite number of values, we approximate by the following sequence of stopping times:

The stopping time takes its values in a finite set and when , . To conclude the proof of the first part of the proposition, it is therefore enough to show that

For this, we are going to prove that the family is uniformly integrable. Let .

Since takes its values in a finite set, by using the martingale property and Jensen’s inequality, it is easily checked that

Therefore, we have

By uniform integrability and convergence in probability, we deduce that

from which it is concluded that

Conversely, let us now assume that for any, almost surely bounded stopping time of the filtration such that , we have .

Let and . By using the stopping time

we are led to

which implies the martingale property for .

The hypothesis that the paths of be right continuous and locally bounded is actually not strictly necessary, however the hypothesis that the stopping time be almost surely bounded is essential, as it is proved in the following exercise.

**Exercise :** * Let be a filtration defined on a probability space and let be a continuous martingale (that is a martingale with continuous paths) with respect to the filtration such that almost surely. For , we denote . Show that is a stopping time of the filtration . Prove that is not almost surely bounded.*

**Exercise :** * Let be a filtration defined on a probability space and let be a continuous martingale with respect to the filtration . By mimicking the proof of Doob’s stopping theorem, show that if and are two almost surely bounded stopping times of the filtration such that and , , then, *

Deduce that the stochastic process is a martingale with respect to the filtration .

**Exercise :** * Let be a filtration defined on a probability space and let be a submartingale with respect to the filtration whose paths are continuous. By mimicking the proof of Doob’s stopping theorem, show that if and are two almost surely bounded stopping times of the filtration such that and , , then, .D*