UoM - Martingales with Applications to Finance: Part I

TOC

1. Probability spaces 1.1 Definition 1.2 Properties of a probability measure 1.3 Sigma field 2. Integration with respect to a probability measure 2.1 F-measurable 2.2 Integration 2.3 Convergence Theorems 3. Conditional Expectation 3.1 Definition 3.2 Important Properties 4. Martingales 4.1 Intro and Definition 4.2 Examples 5. Stopping times 5.1 Definition 5.2 Stopped process 5.3 Doob’s optional stopping theorem 5.4 Doob’s maximum inequality

1. Probability spaces

1.1 Definition

A probability space is a triple , where

is called the sample space consisting of all possible outcomes of the random experiment

is a collection of events (subsets of ), called a -field (collection of information)

is a probability measure which assigns to each event A a number (the probability of A) in such a way that

If is a sequence of disjoint (mutually exlusive) events, then

Example 1

Choose a number randomly from the interval

In this case, . The basic events are sub-intervals . = event that the number we choose is in the interval . Choose so that it contains intervals. For the probability , we set

Example 2

Toss a coin. . is the probability of A. If , then

1.2 Properties of a probability measure

Proposition 1: If is a sequence of events, then

Proof: Let , … , . Then so . Moreover, . Thus

Proposition 2: If , that is, and , then

Proof: Assume . and . Therefore, is disjoint. Thus

Similarly, If , that is, and then as .

1.3 Sigma field

A collection of events (subsets of ) (information) is called a -field if it satisfies the following properties:

if , then

if is a sequence of events belonging to , then also belong to

Examples

The smallest -field:

The biggest -field:

For any event

Def 1.2 For any collection D of events, the smallest -field that contains D is called the -field generated by D, writen as

Example, If , then

2. Integration with respect to a probability measure

Define the integration of a random variable against a probability measure

2.1 F-measurable

A random variable is said to be F-measurable (random variable or F-determined) if the events of form belong to for all choices a, b.

is a funtion. is a random variable. is measurable equals to that

For example: Consider the coin tossing game with 2 rounds

={, , “first coin is head”, “first coin is tail”}. Event A = “second coin is head”. Then is not -measurable. But it is -measurable.

Meaning: is -measurable cotains information of .

Properties:

and are -measurable, then so are

are -measurable, then so are

are -measurable and , then is -measurable

2.2 Integration

Define the in three steps:

1. Integration of a non-negative discrete random variable.

Let be a non-negative discrete random variable with values

In this case, we define that

is called the integral of with respect to .

2. Integration of a non-negative random variable

Let be a non-negative random variable. For , set

Then forms a partition of the sample space , that is . Define the n-th approximation of by

so that

for all . Thus for every , as . Since , it holds that for

Hence . Since is discrete, is already defined as above. Now we define

Note that is allowed in the definition.

3. Integration of random variable (general case)

Let be a given random variable

In this case, we define that

Note that

Define:

if at least on of them is finite.

Definition: Integrable

A random variable is said to be integrable if and are finite.

Since , thus

is integrable

Example: For an event , define

is called the indicator of (non-negative & discrete)

Then

For any event A, define

as the intergration of on the event

Properites of integration

If are integrable, then:

are constants

If , then

If , then . Moreover, if , then almost surely

If and Y is integrable, then is also integrable

2.3 Convergence Theorems

Suppose for each . It is not true in general that .

Example: Let and be the generalized length.

Then for every . However,

The equation can be true if certain conditions are satisfied

Monotone Convergence Theorem

Suppose are random variables with

, and

, then

Example: Let , P is generalized length

Define:

Then clearly, and

By using the Monotone Convergence theorem, we conclude

Dominated Convergence Theorem

Let are random variables such that . In addition, suppose there is a fixed integrable random variable such that for all . Then

Example: Let , P is generalized length.

Define:

From , as .

On the other hand,

Using the Dominant Convergence Theorem,

3. Conditional Expectation

3.1 Definition

Given two events A, B. is the conditional probability of A given B.

Let be a probability space. Let be a -field (collection of events, information). Given two events A, B, we know how to calculate the conditional probability

Define the conditional expectation of given , denoted by , which will be a random variable again regarded as the best estimate of based on the information provided by .

(Definition) A random variable is called the conditional expectation of given , written as , if

is -measurable (determined by )

for any event

This means that the two random variables have the same average (integration) over any event in .

3.2 Important Properties

linear property

If is a random variable determined by (-measurable), then ( can be taken as constant)

In particular,

If is independent of , then

If is another -field (which is smaller than ) such that , then

that is, if take conditional expectation twice, it will end up with the smaller one

Proof of Properties

(2)

Let . Since , we have

(the definition of the conditional expectation)

(4)

Assume is independent of . Let . To prove , we need to check (i) and (ii)

since is a constant, so it’s certainly -measurable (determined by )

For any , we need to show that . Since is independent of , it follows that

(5)

Prove

Let , . We need to show that , check two things:

is -measurable. Since , by definition, is -measurable

check

By definition, .

Since , we have , because , thus .

Prove : because is -measurable, thus the equation is valid.

4. Martingales

4.1 Intro and Definition

Martingales are mathematical models for fair games.

For a family of random variables , denote by the smallest -field containing the events of the form for all choices of .

is called the -field generated by . Random variables determined by are functions of .

Example: Consider a series of games decided by the tosses of a coin, in which we either win $1 with probability or lose $1 with probability in each round. Let denote the net gain in the -th round. Then are independent random variables with

and so

Total net gain after the n-th round is given by and

Let denote the -field generated by ( contains all the information up to round ). Then and can be regarded as the history of the games up to time (the -th round). The average gain after the -th round given the history up to time .

since is determined by (-measurable) and is independent of .

Thus

In the first case, the game is fair, is called a martingale. In the second case, it is called a submartingale. In the third case, it is called a supermartingale.

Definition: A sequence of random variables is said to be a maringale (submartingale, or supermartingale) with respect to an increasing sequence of -fields if :

is determined by (-measurable)

is integrable, i.e.

sequence correlation

Martingale:
Submartingale:
Supermartingale:

Immediate properties

The notion of martingales is mathematical formulation for fair games

If is a martingale, then

The expectation remains constant for all

If is a martingale, then we have

More generally, it holds that for any ,

4.2 Examples

Ex1: Let be independent, integrable random variables with for all . Prove that is a martingale with respect to

Proof:

(i) Since is a function of , it is -determined.

(ii) integrable

(iii) condition 3

So is a martingale.

Ex2: Let be an integrable random variable and be a sequence of increasing -field. Define . Then is a martingale w.r.t

Proof:

(i) Because of the definition of conditional expectation, is -measurable.

(ii) , thus

(iii)

Ex3: If are independent, integrable random variables with for all , then

is a martingale w.r.t.

Proof:

(i) is function of thus is -measurable

(ii) since is integrable and independent thus is integrable.

(iii) Condition 3

Ex4: Let be the net gain process in a series of fair games. Then , and , . We already know that is a martingale w.r.t. . Prove that is also a martingale w.r.t.

Proof:

(i) Since is a function of , is -measurable.

(ii) As , is integrable.

(iii) condition 3

Thus

Ex5: Let be a sequence of independent random variables with . Set . Define . Prove is a martingale w.r.t

Proof:

(i) Because is a function of , is -measurable

(ii) Since , thus

(iii) condition 3

5. Stopping times

Example: Let be the net gain after the n-th round in a series of games. Let be the first time at which net gain reaches 100, i.e.

Such random times are called stopping times.

5.1 Definition

Let be an increasing sequence of -fields. A random variable taking values in the set is called a stopping time (or optional time) w.r.t. if for every n

Example

Let be the net gain process in a series of games. Set

Define

Then is a stopping time w.r.t. .

Indeed, for every

5.2 Stopped process

Let be a stopping time. Let be a sequence of random variables. Set . Define

is called the stopped process of at the stopping time .

Proposition: If is a martingale w.r.t. , then the stopped process is also a martingale (stopped martingale) w.r.t. . In particular,

Proof

Note that

This immediately implies that is -measurable and integrable. It remains to check the property (iii) in the definition of martingale.

That is if , the left is equal to the right, which is . While if , then the left is also equal to the right, which is . Thus

Since , it follows that

Thus is martingale (stopped martingale).

Example

.Is it a stopping game?

{T=3} = {3 is the last time in [0, 100] such that } = {} = {} {} … {} . Thus T is not a stopping time.

5.3 Doob’s optional stopping theorem

Namely, will not be true. For example, in a series of fair game, define , then . However, the following theorem holds.

Theorem: Let be a martingale and an almost surely finite stopping time. In each of the following three cases, we have :

is bounded, i.e. there is an integer such that

The sequence is bounded in the sense that there is a constant such that for all

and the step are bounded, i.e. there is a constant such that for all

Proof

Using the dominated convergence theorem. First of all, as . Since the stopped process is a martingale, we always have

Since , we have and thus

Since is bounded for all , it follows from the dominated convergence theorem that

Write

Consequencely,

By virtue of the dominated convergence theorem, we have

Application of the Theorem

Gambler’s ruin problem

Gambler and play a series of games against each other in which a fair coin is tossed repeatedly. In each game gambler wins or losses $1 according as the toss results in a head or a tail respectively. The initial capital of gambler is $a and that of gambler is $b and they continue playing until one of them is ruined. Determine the probability that will be ruined and also the expected number of games played.

Let be the fortune of gambler after the n-th game. Then

where are independent r.v.’s with ,

The game will stop at the time the gambler or is ruined, i.e. the game stop at

forms a martingale. Since or we have

Note that

Since for all , it follows from the Doob’s theorem that which is

Thus

Next we try to find the expected number of games .

Define . It has been proved that is a martingale. So the stopped process is also a martingale. Thus we have

This yields by convergence theorem that

uses monotone convergence theorem and uses dominated convergence theorem.

Example 1

Write for the independent steps of the walk. Then with

Define . Then as we have seen before is a margingale w.r.t. . Let be the first time at which the random walk hits 0 or . Since , by the Doob’s optional theorem

On the other hand

Therefore,

Example 2

Doubling Strategy: Define the game and the game stop if gain = 1

thus , . Define which means

Is a martingale? Yes

By definition, is -measurable (is the expression of )

thus

Thus is martingale

Is Doob’s Optional Stopping theorem established？ No

(the only circumstance the game ends)

While because is stopped martingale. Why:

is not bounded, that is, can finite

is not bounded. On event , which can go to

is not bounded. Since is not bounded.

Some properties of the game

Almost surely win 1 if playing long enough

Loss can be

5.4 Doob’s maximum inequality

(Theorem.1) Let be a martingale such that for some . Then for every :

and if

That is, for martingales, the probability and the expectation can be controlled by the expectation of

Proof

Let

is a bounded stopping time. Since is a submartingale, we particularly have

Note that

and

We have

This immediately gives

For a real-valued -adapted process and an interval , define a sequence of stopping times as follows:

We set

Then is the number of upcrosssings of for the interval .

(Theorem.2) Let be a supermartingale. We have

Proof

We can assume that the process is stopped at and also . Otherwise consider the process . Set

Then , and on . Note that is the event that the process has at least upcrossings for the interval . Hence, we have

Thus,

Adding the above inequality from to we obtain

(Theorem.3) Let be a martingale such that for some . Then

almost surely.

Proof

For , set

is the number of upcrossings of the processs . By theorem we have

This yields that , where . Set

Then and hence . Since

where denotes te set of rational numbers, we deduce that

Hence

Example 1

Let be a sequence of increasing -fields. Let be an intergrable random variable. Set . Explain why the limit

exists.

Solution: We knew from previous examples that is a martingale. Moreover, we have

By theorem.3 exists.

Example 2

Let be independent random variables with . If and , show converges almost surely.

Solution: Set . It is easy to verify that is a martingale. Furthermore,

It follows from the martingale convergence theorem that

exists. Since by assumption

esists, we conclude that

exists.