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