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Language: en
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Now we are moving into the fourth example.
The fourth example we started with the filtration.
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Fn is a sequence of s-fields and this is the
filtration. Let X be any random variable with
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the random variable is a integrable. Define
Xn is a conditional expectation of X given
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Fn. So you are defining a sequence of random
variables with the help of the conditional
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expectation of the random variable with a
given the information up to n or the filtration
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n.
So suppose you want to prove this sequence
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of random variable or the stochastic process
is a martingale, then it has to satisfy the
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three conditions, which we discussed in the
discrete-time of a martingale property.
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So the first we are checking whether the random
variable Xn is integral. So if you want to
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prove the Xn is a integrable, then you have
to prove the expectation in absolute random
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variable has to be a finite. That is finite.
Then the random variable is integrable.
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So this is same as expectation of in absolute
you can replace Xn by expectation of X given
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Fn. You can take the absolute inside the expectation
and this is nothing but it is an absolute
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of X because that is a way we define since
Xn be any random variable with the expectation
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is a finite and you know the definition of
expectation of expectation X given Y is going
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to be expectation of X. So we are using that
property.
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Hence, expectation of expectation of X given
the filtration Fn is same as expectation of
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that random variable. So here the random variable
is absolute X and this is already proved that
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it is a finite. Therefore, this is also going
to be finite value. Hence, the expectation
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of absolute Xn is finite. The random variable
Xn is integrable. So the first condition is
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verified.
By the definition of conditional expectation
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the, Xn is Fn measurable. The way we have
written Xn is expectation of X given Fn, so
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this is the definition of conditional expectation
whenever you write conditional expectation
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of X given Fn and that exists with the Xn,
that means the random variable Xn's are Fn
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measurable for all n. So by the definition
of conditional expectation, Xn is Fn measurable
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for all n. So the second condition also satisfied.
Now we are going to verify the third condition.
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The expectation of Xn+1 given Fn is same as
you can replace Xn+1 by the definition, that
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is expectation of Xn given Fn+1. You are replacing
Xn+1 with the conditional expectation given
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Fn.
You know the property of the filtration. The
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filtration property is F1 contained in F2,
F2 is contained in F3 and so on. Therefore,
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Fn is contained in Fn+1 if you have two s-fields
and one is the sub-sigma field of other one.
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Fn is a sub s-field. Fn is the sub-sigma fields
of Fn+1. Then the conditional expectation
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of conditional expectation X given Fn+1 given
Fn is same as conditional expectation of X
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given Fn. We are using the conditional expectation
given s-fields with two s-fields Fn contained
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in Fn+1. We are using the property. Hence,
expectation of X given Fn, by the definition,
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expectation of X given Fn is nothing but Xn.
Therefore, this is equal to Xn.
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So left hand side we started with the expectation
of Xn+1 given Fn. The right hand side we land
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up Xn. This is the third property of -- this
is the third condition we have defined it
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in the Martingale property in discrete time.
So, hence, all the three conditions are satisfied
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by the sequence of random variable Xn's. Therefore,
the stochastic process Xn is a Martingale
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with respect to the filtration Fn because
we have used this filtration to conclude Xn
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is a Fn measurable and find out the conditional
expectation is same as Xn and the random variable
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Xn is a integrable. Hence, the stochastic
process is a Martingale with respect to the
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filtration Fn.
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Now we present the example which was introduced
in the beginning of lecture in this module.
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So this is the example we have given as a
motivation for the model Martingale. A player
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plays against an infinitely rich adversary.
He stands to gain rupees 1 with the probability
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p and lose rupees 1 with the probability q.
We are defining the random variable Xn, the
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player's cumulative gain in the first n games.
What will be his fortune, on the average,
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on the next game given that his current fortune?
We are asking the measure in a conditional
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expectation. The game is fair if and only
if this sequence of random variable Xn is
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a Martingale and that is -- this sequence
of random variables or this stochastic process
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will be a martingale when p and q is equal
to 1/2.
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That means when the game is fair, when the
game is fair, that means whether he will gain
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one rupee or he lose one rupee with the equal
probability 1/2 and 1/2, then when the game
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is fair, whenever the game is fair, the given
stochastic process is a martingale and also
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whenever the given stochastic process is a
martingale, in that case, the game is -- the
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game will be a fair game, that means the p
and q will be half, 1/2.
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The conclusion is it says that the player's
expected fortune after one more game played
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with the knowledge of entire past and the
present is exactly equal to his current fortune.
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The conditional expectation of his one more
game expected fortune given the knowledge
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of entire past and present, that means the
filtration till time t or till time n in the
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discrete case, that is same as exactly equal
to his current fortune. That means it is same
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as expected -- that is same as X suffix n
for a discrete case or it is X of t for a
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continuous case.
So whenever the game is fair, that is a p
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and q is equal to 1/2, then the given stochastic
process Xn is a Martingale and the conclusion
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of the -- conclusion of this problem is the
player's expected fortune after one more game
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played with the knowledge of entire past and
present is exactly, exactly is important because
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later we are going to say more than or less
than. For that we are going to name the different.
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So here it is exactly equal to his current
fortune.