Homework 4 due Monday, May 12 at 5 PM.

Lawler Secs. 5.1-5.3References: Karlin and Taylor Ch. 6

Martingales are not so much a stochastic model (though sometimes they can be posed as one) as much as a concept that can be used to clarify models and simplify calculations.

First we need to introduce the concept of a filtration

which is an increasing family of algebras:

Filtrations are a way to characterize information, and in a probabilistic context, one should think about as characterizing the information (of a certain type) available at epoch n.

A typical way to define a filtration is to say that it is generated by a certain stochastic process (or multiple stochastic processes) and then essentially

A discrete-time stochastic process is a martingale with respect

to a filtration

• for all •

provided:

Be careful to note that Markov chains and martingales are distinct concepts; one does not imply the other.

MartingalesFriday, May 02, 20141:57 PM

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does not imply the other.

Continuous-time martingales can be defined analogously, work the same with some more technical complications. We'll stick with discrete-time martingales in this class.

Elementary examples of martingales (Karlin and Taylor Sec. 6.7)

Suppose we are given an independent (not necessarily identically distributed) collection of independent random variables

with .

1)

Then is a martingale with respect to the

filtration generated by the .

Let's show this by checking the two properties: •

(equivalently this means the same as:

Proof:

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Second property of martingale:

Suppose we are given a collection of independent (not necessarily identically distributed) random variables

with and

2)

Claim:

is a martingale w.r.t filtration generated by (call

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random variables with and

Claim:

is a martingale w.r.t filtration generated by (call

it .

(Key observation: Note that martingales often arise from taking a certain combination of random variables and correcting them so they have the martingale property.)

Check both martingale properties:

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The condition can be shown by triangle inequality.

Let's now construct a martingale associated to a branching process. 3)

where the are iid random variables with

Claim is that

is a martingale with respect to the filtration

generated by the branching process

Check the first martingale property:

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Check that by a recursion argument.

Karlin and Taylor have a vast array of other elementary martingale examples. What's the point of making up martingales?

modeling•simplify calculations•

Martingales have two primary practical uses:

In modern financial modeling, martingales are associated with "lack of arbitrage" which is another way to say the market is fair. But of course money just grows based on interest, so one often encodes absence of trading advantage by saying that if is the return on some investment, then is a martingale where r is the interest rate or expected rate of return.

•Modeling: Martingales encode the notion of a "fair game": wealth should on average be preserved.

Martingale convergence theorem: Under rather weak technical conditions, martingales converge. See Karlin and Taylor.

•

Lawler Sec. 5.3, Karlin and Taylor Sec. 6.3-6.5○

Given a martingale with respect to a filtration

(that means that

a Markov time with respect to the same filtration.

Optional Stopping/Sampling Theorem: This theorem is useful for calculation of quantities associated to Markov times (recall absorption probabilities and additive functionals).

•

Simplifying calculations: Martingales are sufficiently special that some powerful theorems apply to them.

and

for all n=0,1,2,…)

• •

•

And suppose the following three conditions are satisfied:

Then .

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