WEBVTT
Kind: captions
Language: en
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So, now we are going to talk about how to
get conditional expectation, when I do not
00:00:19.930 --> 00:00:29.480
have the Sigma-algebra G given by a countable
partition of omega, so how you generalize.
00:00:29.480 --> 00:00:38.280
So, in the case where we had this issue of
countable partition, you know that we proved
00:00:38.280 --> 00:00:45.649
that there would exist a G measureable random
variable E X by G such that integral E X by
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G dP over any A and G is same as integral
of X dP over A, A belonging to G.
00:00:53.820 --> 00:01:06.040
Now this idea, cannot be so easily proved,
if you consider that G is just a Sigma-algebra.
00:01:06.040 --> 00:01:38.869
So, in many books, especially books of finance,
they define the conditional expectation
00:01:38.869 --> 00:01:43.360
X by G as follows.
00:01:43.360 --> 00:02:00.500
So, it is a random variable, is a G measurable
random variable, because that is what we proved
00:02:00.500 --> 00:02:17.909
earlier, so we are assuming it okay, random
variable r.v. And number 2, whatever we have
00:02:17.909 --> 00:02:22.609
proved in the last class, in the last result,
we want to assume that such a thing actually
00:02:22.609 --> 00:02:33.540
happens, as integrating over this and integrating
X are the same thing.
00:02:33.540 --> 00:02:45.439
So, this variable and X does not have much
of a difference, that is what one other crude.
00:02:45.439 --> 00:03:02.249
So, this property is sometimes called partial
averaging. So, this is the definition. So,
00:03:02.249 --> 00:03:05.599
conditional expectation is a G measurable
random variable which satisfies this, which
00:03:05.599 --> 00:03:11.560
shows that, this may not be unique, there
cannot be any more than one G measurable random
00:03:11.560 --> 00:03:18.180
variable which can actually get you this.
But the key question is, whether such a G
00:03:18.180 --> 00:03:20.959
measurable random variable would at all exist.
00:03:20.959 --> 00:03:26.819
Okay, I am making a definition, but try to
generalize my last result, of the last class,
00:03:26.819 --> 00:03:30.510
but how do I know, that this such a thing
such a random variable would exist. If it
00:03:30.510 --> 00:03:36.620
does not exist, then such a thing is nonsense.
To know that it exists, we need to go through
00:03:36.620 --> 00:03:42.680
some steps and get into some slightly deeper
probability theory. We will not push you for
00:03:42.680 --> 00:03:43.680
proofs etc.
00:03:43.680 --> 00:03:48.709
We will try to give you an information about
how things are done what are the thought processes
00:03:48.709 --> 00:03:53.819
behind the steps what that is required or
tools that is required to prove that such
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a random variable would exist which will exhibit
this property, okay.
00:03:57.359 --> 00:04:15.689
One of the, first steps is to know, in finance
and in probability theory is, how to construct,
00:04:15.689 --> 00:04:40.830
a new probability measure from an old one,
that is the key question. So, in that sense
00:04:40.830 --> 00:04:52.930
suppose you have a given probability space,
so whatever handwritten notes I have, I will
00:04:52.930 --> 00:04:59.789
see whether, I mean, I finish delivering the
lectures, I can hand it over to the Muk, support
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and staff, who can actually very good technical
staff here, who can actually scan it and put
00:05:06.050 --> 00:05:12.590
it up on the portal, so that you can see all
these things. So, I cannot write down each
00:05:12.590 --> 00:05:16.190
and everything in detail which is written
in the notes because I am just trying to explaining
00:05:16.190 --> 00:05:17.190
the real issues.
00:05:17.190 --> 00:05:24.199
So, here we take a probability space and we
take a random variable and assume that this
00:05:24.199 --> 00:05:32.319
random variable is non-negative. You can assume
it, in the sense of almost everywhere but
00:05:32.319 --> 00:05:37.620
okay let us for the moment assume that, X
omega is greater than 0 for every instance
00:05:37.620 --> 00:05:54.970
omega. Now, also assume, you have X is equal
to 1.
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You see, once I know this, then I construct
for A in the Sigma-algebra, if A is a event
00:06:05.360 --> 00:06:25.070
then, construct the following function of
A. This Q, which is actually taking an element
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of a Sigma-algebra and mapping to R given
this is integrable. So, suppose that X is
00:06:34.550 --> 00:06:52.220
integrable actually. Then this Q generates
a new probability measure. So, then Q okay
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maybe I will write it like this no maybe I
will not do this because that will make it
00:07:01.080 --> 00:07:23.639
look like rational numbers. So, this Q is
a probability space, that is absolutely fundamental.
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So, given a probability measure, I can construct
a new probability measure in the following
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way.
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So, another interesting property that we have
is that, if Y is a random variable on
00:07:56.909 --> 00:08:10.240
and suppose I can integrate, then for any
A that you have in F, so this is nothing but,
00:08:10.240 --> 00:08:21.800
so this is an additional result. So, this
is the key step, where you have constructed
00:08:21.800 --> 00:08:23.909
the new probability measure.
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Finance is very important, how to move from
one probability measure to the other. An important
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result that can be shown is that, suppose
P of A
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is 0 okay. Then, in this particular case,
it would imply that Q of A is equal to 0.
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This is called Q have been absolutely continuous
with respect to P. So, if you have an A for
00:09:10.530 --> 00:09:18.920
which P of A is 0, then Q of A is also equal
to 0. This is fundamental which comes out
00:09:18.920 --> 00:09:19.920
from here, this definition.
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So, here you see Q, now how do I know that
Q is a probability measure; because you see,
00:09:35.850 --> 00:09:45.680
Q of omega is integral omega X dP and this
is nothing but expectation of X, which I have
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taken to be 1, so probability of the sure
event omega that whole sample space must be
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1. This is something very important that,
it is truly a probability measure and you
00:09:59.100 --> 00:10:09.529
can show that this will happen. So, the interesting
question that we now can ask comes from here.
00:10:09.529 --> 00:10:19.259
So, if I have a situation where P is the probability
measure and Q is the probability measure on
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both on so P Q are both probability measures
on omega F. So, we have this probability space
00:10:29.230 --> 00:10:39.760
and we have this probability space also right.
So, this probability space and this probability
00:10:39.760 --> 00:10:50.791
space, but we also have this connection that
P A equal to 0 would imply Q A equal to 0.
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Whenever P A is 0 Q A is 0. Then is Q, a probability
measure constructed out of P. So, which means
00:11:02.160 --> 00:11:12.449
the question is can Q be constructed out of
P?
00:11:12.449 --> 00:11:25.680
Answering this question leads to a deep result
called the Radon-Nikodym theorem and the random
00:11:25.680 --> 00:11:30.779
variable which will require to construct Q
in this manner, the one which we have already
00:11:30.779 --> 00:11:36.320
shown in this result, that random variable
would be called the Radon-Nikodym derivative
00:11:36.320 --> 00:11:40.949
and this is a very-very famous result in measured
theory and in probability, also it is called
00:11:40.949 --> 00:11:47.279
the Radon-Nikodym theorem. Just take this
for information do not get too worked up because
00:11:47.279 --> 00:11:52.670
you are not going to really use Radon-Nikodym
theorem in such a way in your actual practice
00:11:52.670 --> 00:12:02.279
of finance, but it is good to have some additional
information.
00:12:02.279 --> 00:12:11.990
So, what does Radon-Nikodym theorem tell me?
Radon-Nikodym theorem tell me the following,
00:12:11.990 --> 00:12:26.660
tells me that okay, very good, so you have
this information. If P A equal to 0 implies
00:12:26.660 --> 00:12:39.110
Q A equal to 0, where A is of course element
of F and A is element of F, A must be an event
00:12:39.110 --> 00:12:54.329
of course you do not compute the probability,
then there exists a random variable Y which
00:12:54.329 --> 00:13:00.290
is nonnegative, an integrable.
00:13:00.290 --> 00:13:11.400
This is a very deep result because we are
proving the existence of something. Every
00:13:11.400 --> 00:13:16.199
existence here results in mathematics as deep
results. It is not easy to prove that something
00:13:16.199 --> 00:13:21.339
actually exists whether you are able to really
hand calculate it that is a different matter
00:13:21.339 --> 00:13:26.550
but the question of its existence is a very-very
important issue in the way modern mathematical
00:13:26.550 --> 00:13:27.740
thinking goes.
00:13:27.740 --> 00:13:34.480
So, you are telling such a thing would actually
be there and so then there exists random variable
00:13:34.480 --> 00:13:41.519
this which is integrable such that, Q of A
can be constructed in the following manner.
00:13:41.519 --> 00:13:45.461
See, why you need to talk about a positive
random variable not just any random variable
00:13:45.461 --> 00:13:47.300
taking any value.
00:13:47.300 --> 00:13:54.209
You need to talk about a positive random variable
whether it is here or here because then you
00:13:54.209 --> 00:14:00.839
will always have Q A greater than equal to
0. You put five here it would become 0. It
00:14:00.839 --> 00:14:05.720
will put omega here it is 1 so that is a true
probability measure. Q A values are lying
00:14:05.720 --> 00:14:13.640
between 0 and 1 because X is nonnegative this
will always be greater than equal to 0 and
00:14:13.640 --> 00:14:20.759
that is why you need to talk about our nonnegative
random variable, okay.
00:14:20.759 --> 00:14:25.069
So, you will always get this. There would
always exist a nonnegative random variable
00:14:25.069 --> 00:14:31.240
such that this will happen. Of course, E of
Y would be 1 because, you already now know
00:14:31.240 --> 00:14:42.490
that, Q is the probability measure so Q of
omega is 1 so expectation of Y is 1. So, let
00:14:42.490 --> 00:14:54.110
now interesting fact is, that this is unique
up to a measure, means it says that if Z greater
00:14:54.110 --> 00:15:04.050
than equal to 0 is a random variable, such
that Q of A can also be written like this,
00:15:04.050 --> 00:15:15.089
for every A element of F. Then it implies
that Y and equal to Z almost surely that Y
00:15:15.089 --> 00:15:22.829
and Z agree at every omega except on a set
of omega of measure 0.
00:15:22.829 --> 00:15:30.589
So, this is essentially unique so this Y is
usually written as dQ dP the Radon-Nikodym
00:15:30.589 --> 00:15:38.490
derivative of Q with respect to Q given P
and Q are absolutely continuous measure means
00:15:38.490 --> 00:15:43.500
Q is absolutely continuous with respect to
P.
00:15:43.500 --> 00:15:52.379
So, this is the Radon-Nikodym theorem which
tells you that okay if you can construct a
00:15:52.379 --> 00:15:57.540
probability measure if you have random variable
like this you can always construct a probability
00:15:57.540 --> 00:16:01.240
measure, for which this property will hold.
And the reverse says that if this property
00:16:01.240 --> 00:16:07.819
hold this will always get a nonnegative random
variable, for which would be able to through
00:16:07.819 --> 00:16:12.460
which you are able to represent the measure
Q, in terms of the probability measure Q,
00:16:12.460 --> 00:16:17.430
in terms of the probability measure P.
00:16:17.430 --> 00:16:26.029
But these two things sum up to give you, there
is no time to really write down proofs, but
00:16:26.029 --> 00:16:31.920
I am trying to tell you that this two results
these two ideas sum up to prove the existence
00:16:31.920 --> 00:16:41.100
of a random variable, which will satisfy exactly
what we have seen in the definition of conditional
00:16:41.100 --> 00:16:44.480
expectation in the general setup.
00:16:44.480 --> 00:17:22.250
So, existence of conditional expectation.
So, what does this result say. So, let us
00:17:22.250 --> 00:17:38.650
start with this fact that, X be an integrable
random variable on the probability space,
00:17:38.650 --> 00:17:49.340
okay and G subset of F is a sub sigma field
okay otherwise we ll just say this sigma field
00:17:49.340 --> 00:18:03.940
or Sigma-algebra whatever term you want to
use you can use, Sigma-algebra on, then there
00:18:03.940 --> 00:18:14.650
exists a random variable E X by G on.
00:18:14.650 --> 00:18:26.260
So, this is an important result, which is
integrable. So this sign is very common to
00:18:26.260 --> 00:18:34.520
mathematician, this means there exists. So,
then there exists a random variable on this,
00:18:34.520 --> 00:18:41.210
on the probability space such that, the partial
averaging property, which we had written down
00:18:41.210 --> 00:19:04.020
actually holds. So, it shows, that though
this random variable may not be unique, but
00:19:04.020 --> 00:19:08.450
such a random variable do exists and exhibits
this partial averaging property. So, this
00:19:08.450 --> 00:19:16.130
is true for all A you take in Sigma-algebra
G with respect to which you are doing the
00:19:16.130 --> 00:19:17.130
condition.
00:19:17.130 --> 00:19:31.150
Now the interesting part is the following.
So, suppose Y is a random variable on this
00:19:31.150 --> 00:19:44.060
probability space, omega F P and integral
A Y dP is equal to integral X dP over A, for
00:19:44.060 --> 00:19:55.030
all A in G then it implies that, Y is equal
to the conditional expectation of X Event
00:19:55.030 --> 00:20:04.740
G almost surely or almost everywhere. So,
any random variable Y which satisfies this
00:20:04.740 --> 00:20:09.810
property must be same as the conditional expectation
that is the fundamental idea.
00:20:09.810 --> 00:20:15.100
So, that also shows that this is really not
unique you can just find some Y which satisfies
00:20:15.100 --> 00:20:23.010
this, that is, which is G measurable. So of
course, Y has to be G measurable. So, if you
00:20:23.010 --> 00:20:34.260
can find any G measurable function this will
just work.
00:20:34.260 --> 00:20:41.070
Now, I will state the properties of conditional
expectation. I will not give the proof of
00:20:41.070 --> 00:20:46.300
the properties. The proof of the properties
would be coming through exercises. You will
00:20:46.300 --> 00:20:53.150
be asked to give a proof of the properties
and then the proof would be given in your
00:20:53.150 --> 00:20:59.390
homework. Maybe if there is time, time permits
I will try to do for 1 and not each and everything.
00:20:59.390 --> 00:21:05.290
So, let me write down the properties of conditional
expectation. So, these are important properties
00:21:05.290 --> 00:21:16.260
used very often in finance so it is important
to know this.
00:21:16.260 --> 00:21:37.560
Properties of conditional expectation. So,
what is the first property? First property
00:21:37.560 --> 00:21:50.360
is taking out what is known. Number 1 property,
is called the taking out what is known property.
00:21:50.360 --> 00:21:57.810
So, what do I mean by this.
00:21:57.810 --> 00:22:04.700
So, suppose I have 2 random variables X and
Y and I say, that X and X into Y that is X
00:22:04.700 --> 00:22:20.280
omega into Y omega for the omegas, when you
compute them is, integral, is integrable and
00:22:20.280 --> 00:22:41.050
X is G measurable, means, G contains all information
about X that is the meaning. So, that is the
00:22:41.050 --> 00:22:46.450
meaning of something is known. So, X is essentially
known by knowing G. So, the nature of X is
00:22:46.450 --> 00:23:04.930
completely revealed to G. So, this is nothing
but X into expectation of, this is the thing.
00:23:04.930 --> 00:23:20.520
Number 2, expectation of expectation of X
by G is expectation of X. This is a very simple
00:23:20.520 --> 00:23:27.360
thing, this property because, if you use the
partial averaging property, what it means.
00:23:27.360 --> 00:23:40.800
What does expectation of a thing mean? Integral
omega expectation of X by G, this means this,
00:23:40.800 --> 00:23:50.500
and this means omega X dP and that is expectation
of X. So, this is the partial averaging property
00:23:50.500 --> 00:23:56.780
and this nothing, it gives me this because
these 2 for any A in G and of course omega
00:23:56.780 --> 00:23:59.540
the whole space A is in G because G is a Sigma-algebra.
00:23:59.540 --> 00:24:05.000
So, as I have told you in the very beginning
that the whole omega has to be a null event
00:24:05.000 --> 00:24:09.670
and sure event has to be the whole sample
space has to be in any sigma fields. So, that
00:24:09.670 --> 00:24:17.730
is it and this is nothing but the expectation
of expectation of X by G. So, you have taken
00:24:17.730 --> 00:24:21.680
the expectation of this random variable which
is this and the partial averaging do this
00:24:21.680 --> 00:24:26.650
which gives you this.
00:24:26.650 --> 00:24:31.520
This is property two then we come to property
three and property four.
00:24:31.520 --> 00:24:42.320
Property three is called independence. So,
if X and G are independent, I will explain
00:24:42.320 --> 00:24:50.410
to you, what I mean, if X and G are independent,
you will be very surprised that I am talking
00:24:50.410 --> 00:24:54.410
about independence of a random variable with
respect to a Sigma-algebra. I will tell you
00:24:54.410 --> 00:25:03.020
what is that.
So, mathematicians hardly mix-up lot of language.
00:25:03.020 --> 00:25:11.880
So, if X and G are independent then basically
X does not depend on G so the conditional
00:25:11.880 --> 00:25:16.630
expectation of G is basically a constant random
variable; for any omega that you take this
00:25:16.630 --> 00:25:25.730
gives you the values E X. What I mean by X
and G been independent. By the term X and
00:25:25.730 --> 00:25:34.820
G been independent, I mean that if you take
any A element of G, how do I identify the
00:25:34.820 --> 00:25:40.660
element A, how do I say that the event A has
occurred. Event A, if I take, I look at the
00:25:40.660 --> 00:25:44.390
outcome of random experiments some omega has
occurred in the sample space. If that omega
00:25:44.390 --> 00:25:48.050
remains is in A then I say that the event
A has occurred.
00:25:48.050 --> 00:25:52.760
For example, I throw a die and the event A
is that an odd number occurs. If I say that
00:25:52.760 --> 00:25:59.710
the number 3 has occurred, then I say that
the event A has occurred. So, which means,
00:25:59.710 --> 00:26:04.230
this event A, can be identified in terms of
a random variable I A the indicator function.
00:26:04.230 --> 00:26:12.360
So, where IA is a function defined as follows
where I A omega is equal to 1, if omega is
00:26:12.360 --> 00:26:19.080
in A and is equal to 0, if omega is not in
A.
00:26:19.080 --> 00:26:32.750
So, which means that X and I A where A is
element of G this are independent. So, this
00:26:32.750 --> 00:26:37.080
are for any A and G are independent random
variables. So, this is a random variable,
00:26:37.080 --> 00:26:46.990
independent random variables. That
is the meaning of the fact that X and G are
00:26:46.990 --> 00:26:53.140
independent. So, if these are independent,
so X really does not depend upon the occurrence
00:26:53.140 --> 00:26:57.840
of any event in G then of course this is the
case.
00:26:57.840 --> 00:27:11.650
Third and the last property which is used
pretty often is a tower property. So, it is
00:27:11.650 --> 00:27:19.660
very important for you to learn these properties.
So, you will have exercises to work on. So,
00:27:19.660 --> 00:27:28.350
what do you do. You have a sigma field H,
which is contained in the sigma field G, which
00:27:28.350 --> 00:27:40.750
is contained in the sigma field F, so this
both sorry, so both G and H are sigma fields,
00:27:40.750 --> 00:27:44.520
which is contained in the sigma field H.
00:27:44.520 --> 00:28:02.630
Then the property that we have is the following
that, expectation
00:28:02.630 --> 00:28:18.150
of X by G given H, is same as expectation.
So, conditional expectation of the random
00:28:18.150 --> 00:28:24.630
variable E X by G given that H has occurred
is same as sorry, E X by H.
00:28:24.630 --> 00:28:31.700
So, this is something I would like you to
remember. So, these are four properties that
00:28:31.700 --> 00:28:37.450
you learn about conditional expectation are
just very crucial. So, lot of things you have
00:28:37.450 --> 00:28:41.470
to use repeatedly when you study finance and
when you start studying Martingales.
00:28:41.470 --> 00:28:47.300
Martingales are something used for when it
comes term actually comes out with gambling,
00:28:47.300 --> 00:28:53.450
so you have gambling and then as time passes
more information is revealed and if you have
00:28:53.450 --> 00:28:59.220
some information now, what is your expected
gain or profit or expected income or expected,
00:28:59.220 --> 00:29:06.020
whatever you want to say, expected payoff
at the next time. So, that is what Martingales
00:29:06.020 --> 00:29:08.420
talks about and that is what we are going
to learn in the next class.
00:29:08.420 --> 00:29:12.270
It is very interesting and hope because at
the end stock market is gambling. At the end
00:29:12.270 --> 00:29:17.080
the stock market keeping money in the stock
market is gambling, but people do keep money
00:29:17.080 --> 00:29:22.670
in the stock market because, the rate at which
money grows in the stock market if everything
00:29:22.670 --> 00:29:26.600
goes on, is much faster than if you keep it
in the money market which is fixed deposit
00:29:26.600 --> 00:29:29.590
like or invest in government bonds.
00:29:29.590 --> 00:29:36.070
So, it is very important that you have to
have an understanding of this. If you really
00:29:36.070 --> 00:29:40.690
want to know in the second part of the course
how derivatives are priced, of course some
00:29:40.690 --> 00:29:45.490
derivative pricing will, of course come in
here. We will talk about at the end when we
00:29:45.490 --> 00:29:47.760
learn it to calculus. We will learn to apply
it to calculus. We will talk about how to
00:29:47.760 --> 00:29:54.210
compute derivative rates. We will talk about,
what is the actual behavior of the stock prices,
00:29:54.210 --> 00:29:58.520
a pretty good model called the geometric Brownian
motion.
00:29:58.520 --> 00:30:02.890
All these things will come gradually, but
these properties will become pivotal there
00:30:02.890 --> 00:30:10.220
and as a result, I would like you to remember
and use these properties. Do these exercises
00:30:10.220 --> 00:30:15.310
which is associated with this conditional
expectation in a pretty serious way. At least
00:30:15.310 --> 00:30:23.600
learn this part very well. Of course, the
internet is there. You can keep on looking
00:30:23.600 --> 00:30:31.201
at it but here the course is basically, to
give you some understanding. Maybe you will
00:30:31.201 --> 00:30:35.860
do slightly better than the internet, but
anyway, you can use whatever resources you
00:30:35.860 --> 00:30:41.740
want but these things have to be known very
well.
00:30:41.740 --> 00:30:45.200
Thank you very much.