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Language: en
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Okay so we are going to talk about Martingales
today. So what are Martingales? We cannot
00:00:22.700 --> 00:00:34.960
immediately approach that Martingales are
particular type of stochastic processes because
00:00:34.960 --> 00:00:41.250
stochastic process behaves in a certain way,
we will call it Martingale.
00:00:41.250 --> 00:00:56.300
In order to understand martingales, we need
to first talk about the notion of a filtration.
00:00:56.300 --> 00:01:04.110
Of course in any discussion that we begin
with always keep in mind that even if I do
00:01:04.110 --> 00:01:17.830
not mention it underlying everything is the
following probability space okay. Now what
00:01:17.830 --> 00:01:29.300
essentially is filtration? Filtration is the
part what happens is that as a random experiment
00:01:29.300 --> 00:01:36.790
progresses and new information becomes available
you know which part of the Sigma-algebra you
00:01:36.790 --> 00:01:38.450
already know.
00:01:38.450 --> 00:01:43.940
So some part of the Sigma-algebra would be
completely revealed. So some part of F or
00:01:43.940 --> 00:01:53.580
subset of F would be revealed once we start
getting more information. For example I throw
00:01:53.580 --> 00:02:10.530
a fair coin. So if I throw a fair coin, there
are 8 possibilities okay. So let me write
00:02:10.530 --> 00:02:16.299
down what are the 8 possibilities.
00:02:16.299 --> 00:02:27.269
This is 1 or we can have all heads; head,
tail, and then head; head, tail and tail;
00:02:27.269 --> 00:02:41.790
tail, tail, and head; tail, head, and head;
tail, head, tail; and tail, tail, tail. These
00:02:41.790 --> 00:02:52.870
are the 8 possibilities if you throw 3 coins
or 1 coin thrice in succession. So you are
00:02:52.870 --> 00:03:03.220
doing a repeated experiment. Now suppose I
know that the first coin I tell you okay suppose
00:03:03.220 --> 00:03:07.870
I do not allow you to see the experiment,
I am conducting the experiment and I tell
00:03:07.870 --> 00:03:21.650
you okay the first coin has turned out to
be head. Then what are revealed to you?
00:03:21.650 --> 00:03:31.250
What events are revealed to you? If the first
coin is head
00:03:31.250 --> 00:03:41.180
is the situation is this one, means you know
if the first coin is head the only possibilities
00:03:41.180 --> 00:04:02.489
now are the following sorry. So once I know
that the first coin is head there are only
00:04:02.489 --> 00:04:17.709
these possibilities that can occur or maybe
if the first coin is tail so if the first
00:04:17.709 --> 00:04:22.430
coin is head then these are the possibilities
that can occur. If the first coin comes out
00:04:22.430 --> 00:04:24.610
to be tail then these are the possibilities.
00:04:24.610 --> 00:04:45.650
So if the first coin, the outcome of the first
toss is known to me, then given any omega,
00:04:45.650 --> 00:04:55.509
if you give me any omega in any sequence of
3 tosses, I can tell you whether this will
00:04:55.509 --> 00:05:00.180
be revealed or this would not be revealed,
this will be an outcome or this will be not
00:05:00.180 --> 00:05:05.879
an outcome. So what I tell you okay the first
coin has come out to be head then you say
00:05:05.879 --> 00:05:10.389
okay can tail tail tail come, no. the tail
tail tail come will come in the compliment
00:05:10.389 --> 00:05:15.040
of this H this is the compliment.
00:05:15.040 --> 00:05:22.720
So what we have done? We have essentially
segregated this thing, separated this out.
00:05:22.720 --> 00:05:30.680
Made a finer division of these 2. So given
any omega now, any omega does not belong to
00:05:30.680 --> 00:05:40.870
the empty set, every omega is belonging to
the whole sample space which means the sample
00:05:40.870 --> 00:05:45.850
space and the empty set is always revealed.
You know that either there will be nothing
00:05:45.850 --> 00:05:48.960
or there will be everything basically.
00:05:48.960 --> 00:05:54.699
But the interesting part is that now given
an omega I can tell you whether that event
00:05:54.699 --> 00:06:02.150
will now occur or will not occur. I have the
information. If you say first toss is actually
00:06:02.150 --> 00:06:08.340
head, I can tell you what will actually happen,
what are the next consequences any of those
00:06:08.340 --> 00:06:16.120
4, right. If you say okay what about T H T
can this consequence will be there no, it
00:06:16.120 --> 00:06:17.190
cannot be.
00:06:17.190 --> 00:06:25.580
So if you look at it the following sets of
the Sigma-algebra F is now revealed which
00:06:25.580 --> 00:06:38.710
I called F1 which is, see if I know I have
a knowledge about the first toss. These are
00:06:38.710 --> 00:06:44.720
the sets of the Sigma-algebra which is revealed
and this itself F1 itself is a Sigma-algebra
00:06:44.720 --> 00:06:54.169
whether it follows all the rules of the Sigma-algebra
whether if you take the union of these 2 it
00:06:54.169 --> 00:06:55.169
will become omega.
00:06:55.169 --> 00:07:00.789
Now if you take the intersection it will become
phi. If you take the compliment of H it is
00:07:00.789 --> 00:07:09.379
A T compliment of A T is H. So this is a Sigma-algebra.
So this part of the Sigma-algebra, so this
00:07:09.379 --> 00:07:17.580
is of course you immediately see that F1 is
a subset of the Sigma-algebra F. So part of
00:07:17.580 --> 00:07:22.280
the Sigma-algebra gets revealed when some
information is revealed and the next stage
00:07:22.280 --> 00:07:27.680
I said okay good I tell you what has happened
in the second toss.
00:07:27.680 --> 00:07:37.419
Then I will have finer revealing. I can again
partition this into finer parts when basically
00:07:37.419 --> 00:07:44.400
I am breaking up the space F the Sigma-algebra
F.
00:07:44.400 --> 00:07:53.930
So you tell me that so what can be the second
toss either both can be head first head and
00:07:53.930 --> 00:08:02.009
then can be tail or the first can be tail
the second can be head, the first can be tail
00:08:02.009 --> 00:08:09.840
and the second can be tail. So if I tell you
what are the 2 consecutive things you know
00:08:09.840 --> 00:08:15.919
what can occur now. If I have the knowledge
of what is also the second outcome and also
00:08:15.919 --> 00:08:20.409
I know both the outcomes I know what are the
occurrences.
00:08:20.409 --> 00:08:34.820
So here it will be this, nothing but just
augmenting with head or tail that is all.
00:08:34.820 --> 00:08:59.850
So you have made more finer divisions of this
basically. So I know if you said that the
00:08:59.850 --> 00:09:06.519
first coin is head and the second coin is
head if you said the second coin is head so
00:09:06.519 --> 00:09:12.029
these are the 2 things that can happen, anyone
of the things can come. Second coin is head
00:09:12.029 --> 00:09:18.209
because the first coin is could be head could
be tail. So any one of these 2 things can
00:09:18.209 --> 00:09:19.680
come.
00:09:19.680 --> 00:09:29.060
Now you see can I now make some Sigma-algebra
out of this information. So this sigma that
00:09:29.060 --> 00:09:37.480
sigma here for example this Sigma-algebra
totally encodes the information that the first
00:09:37.480 --> 00:09:44.260
toss is known. If I know the second toss,
can a Sigma-algebra be constructed which can
00:09:44.260 --> 00:09:51.680
encode all the information. So of course you
should have once if you want to put construct
00:09:51.680 --> 00:09:56.980
a Sigma-algebra if they are all inside that
Sigma-algebra then all this has to be also
00:09:56.980 --> 00:10:03.010
part of the Sigma-algebra because A C H H
is not any one of them but this whole the
00:10:03.010 --> 00:10:06.610
union basically.
00:10:06.610 --> 00:10:10.430
So you basically do not have to when you construct
the set you do not have to write the union
00:10:10.430 --> 00:10:16.240
of this union of this union of this 3 or union
of any of the 3 because that is the compliment
00:10:16.240 --> 00:10:33.440
of the remaining. So you can take the compliment.
So that is it. So you take the compliment
00:10:33.440 --> 00:10:42.089
and that is so you take the compliment. Of
course you have to take some unions also.
00:10:42.089 --> 00:10:59.240
For example if I take this union such a union
does not such a union for example does not
00:10:59.240 --> 00:11:09.630
appear in any one of these sets already known.
00:11:09.630 --> 00:11:16.360
For example if you take this set and this
set and take their union there is no where
00:11:16.360 --> 00:11:24.490
I can find anything right. But if you take
this set and this set and take their union
00:11:24.490 --> 00:11:33.360
and this set is A T. So I do not need to bother
about the union of these set but I can I have
00:11:33.360 --> 00:11:37.740
to bother about the union of these 2 sets,
union of these 2 sets which were not there
00:11:37.740 --> 00:11:39.660
to create the Sigma-algebra.
00:11:39.660 --> 00:11:46.199
You see H and A T is anyway revealed even
if I know the second choices. The first choice
00:11:46.199 --> 00:11:51.640
must be either head or tail so these are already
there. So whatever is known at the first stage
00:11:51.640 --> 00:11:59.050
will always be carried on to the second stage
because they are anyway revealed. For example
00:11:59.050 --> 00:12:05.949
if I take the union of these 2 and take the
compliment of that that would anyway give
00:12:05.949 --> 00:12:12.500
me A H. So A H and A T anyway will continue
to be revealed right.
00:12:12.500 --> 00:12:24.420
So essentially if I want to construct F2 you
see how far the size of the cardinality of
00:12:24.420 --> 00:12:44.240
this set will grow. So A H and A T will anyway
be there okay. You will also have these sets
00:12:44.240 --> 00:13:21.680
A HH, A HT, A TH, A TT. Of course then you
have to line up their compliments okay. Now
00:13:21.680 --> 00:13:27.410
you can combine this with this. Do not combine
this with this because this with this will
00:13:27.410 --> 00:13:33.040
give you A H so this is already there so you
do not have to show that combination.
00:13:33.040 --> 00:13:51.019
So A HH union A TH must be there must be considered.
A HH union A TT must be considered because
00:13:51.019 --> 00:14:06.269
they would not form anything which is already
given. A HT union A TT must be A TH must be
00:14:06.269 --> 00:14:14.560
considered these 2 and A HT union A TT must
be considered means those things which do
00:14:14.560 --> 00:14:22.820
not appear at all. You see your from just
4 elements here we have increased to 1, 2,
00:14:22.820 --> 00:14:35.440
3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 sorry
1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15,
00:14:35.440 --> 00:14:39.500
16, 17, 18 just blowed up actually.
00:14:39.500 --> 00:14:49.670
Now it comes F3, all is known. All is known,
means F3 means I know all possibilities so
00:14:49.670 --> 00:14:56.110
I basically know the whole Sigma-algebra F.
So F3 and the third if I know all the coin
00:14:56.110 --> 00:15:06.480
tosses are known and F3 is nothing but F and
that will have 256 possibilities, 2 to the
00:15:06.480 --> 00:15:14.120
power 8 combinations because there are 8 possible
elements in the Sigma-algebra.
00:15:14.120 --> 00:15:22.440
So what I have, now let us talk about what
is F 0 then. F 0 means when nothing is revealed
00:15:22.440 --> 00:15:29.820
to me. Then I have only 2 choices. Either
nothing will happen or every possible choice
00:15:29.820 --> 00:15:42.389
might happen right. So F 0 consist of just
so this is under no information. This is under
00:15:42.389 --> 00:15:51.089
1 information that first toss is something.
This is under more information the second
00:15:51.089 --> 00:15:59.140
toss is revealed and this is under the third
we have revealed everything.
00:15:59.140 --> 00:16:05.399
So everything is revealed so which means if
you look at them you have F of 0 belonging
00:16:05.399 --> 00:16:16.620
to F of 1 belonging to F of 2 belonging to
F of 3 belonging to F where F of 3 is equal
00:16:16.620 --> 00:16:26.740
to F you do not write have to write F of 3
subset of F or equal to F. So what I have
00:16:26.740 --> 00:16:34.079
done is a chain of Sigma-algebra each containing
the other. So as time evolves more information
00:16:34.079 --> 00:16:39.370
is revealed, you know more about the structure
of the Sigma-algebra.
00:16:39.370 --> 00:16:47.529
So such a sequence of Sigma-algebras which
each of them are subset of the original Sigma-algebra
00:16:47.529 --> 00:16:53.740
and each of them are bigger than the previous
one, such a thing is called a filtration which
00:16:53.740 --> 00:16:56.439
we can obviously formally define.
00:16:56.439 --> 00:17:10.430
So this is in a very discrete setting. You
can define a filtration in the following way.
00:17:10.430 --> 00:17:19.720
So this definition I have given from the book
of Shreve, Steven Shreveâ€™s book called Stochastic
00:17:19.720 --> 00:17:50.790
Calculus for Finance okay. Now observe what
I want to say. So if you have sigma F P let
00:17:50.790 --> 00:18:11.480
T be a number greater than 0. Assume that
for T greater than equal to 0 less than equal
00:18:11.480 --> 00:18:26.180
to T there exists, this is the symbol of there
exists, a Sigma-algebra F T.
00:18:26.180 --> 00:18:31.040
So when we are going to write continuous things
we will put it F T instead of this T just
00:18:31.040 --> 00:18:34.960
to differentiate between the discrete thing
and the continuous event. When we write the
00:18:34.960 --> 00:18:39.850
discrete thing we will write it as a lowered
index and when we write the continuous thing
00:18:39.850 --> 00:18:42.680
it is as if it is a function.
00:18:42.680 --> 00:19:01.290
So there exists a Sigma-algebra F T such that
whenever I have S less than equal to T, F
00:19:01.290 --> 00:19:24.720
of S must be contained F T such that F T is
contained in F and whenever S is less than
00:19:24.720 --> 00:19:39.520
equal to T will F of S would be contained
in F of T say F of T will contain in F of
00:19:39.520 --> 00:19:51.890
S. So this condition has to satisfy it. Then
if these are all satisfied, then the family
00:19:51.890 --> 00:20:20.180
of Sigma-algebras F T where T is from 0 to
T is called a filtration associated with
00:20:20.180 --> 00:20:34.300
the probability space.
So it might, the definition might look tricky,
00:20:34.300 --> 00:20:43.130
but I think you can go and read it up in books
or just think about it a little bit just think
00:20:43.130 --> 00:20:54.250
about the definition you can if you want rerun
the whole lecture. So you can see it how many
00:20:54.250 --> 00:21:00.510
times you want so that you get your concepts
cleared once for all. So then we are going
00:21:00.510 --> 00:21:10.470
to talk about something called a stochastic
process adapted to the filtration.
00:21:10.470 --> 00:21:21.011
So given a stochastic process X t say is of
this form, so this is the stochastic process
00:21:21.011 --> 00:21:53.780
given to you. We say that this X t is adapted
to the filtration
00:21:53.780 --> 00:22:14.130
F t
if X t is F t measurable so all these are
00:22:14.130 --> 00:22:23.330
these are all random variables. For every
t between 0 to t, X t is some random variable;
00:22:23.330 --> 00:22:38.540
adapted to the filtration if X t is F t measurable
and X t measurable that is X t inverse of
00:22:38.540 --> 00:23:09.030
B belongs to F t for all Borel set B in R
okay. This is the meaning of an adapted stochastic
00:23:09.030 --> 00:23:10.040
process.
00:23:10.040 --> 00:23:16.140
Martingales are special type of adapted stochastic
process. So first we will talk about discrete
00:23:16.140 --> 00:23:20.880
Martingales and then we will talk about continuous
Martingales. Do not get too much bothered
00:23:20.880 --> 00:23:27.490
about their properties right now except the
one which we will require.
00:23:27.490 --> 00:23:43.820
So suppose you have a discrete filtration
say F n, so 0 to say m equal to 0 to capital
00:23:43.820 --> 00:23:53.070
N why I am using a finite class because I
am essentially talking about finance. In finance
00:23:53.070 --> 00:24:01.280
you have a trading horizon, trading time,
so trading starts at 0 and ends at T so ends
00:24:01.280 --> 00:24:06.770
at time N. So that is why I am taking the
simplest definition. You can possibly take
00:24:06.770 --> 00:24:11.750
N equal to 0 to infinity it does not matter
and you can have a stochastic process which
00:24:11.750 --> 00:24:12.750
is infinite.
00:24:12.750 --> 00:24:16.800
Stochastic process are nothing but sequences
of random variables where the indexing is
00:24:16.800 --> 00:24:45.110
actually overtime. F n is a given filtration.
And then you have X n, n equal to 0 to capital
00:24:45.110 --> 00:24:53.540
N, a stochastic process adapted to this filtration.
So you will hear this term adapted to filtration
00:24:53.540 --> 00:25:33.830
repeatedly after this. Stochastic process
adapted to the filtration F n. So now we will
00:25:33.830 --> 00:26:06.090
call X n to be a distinct Martingale
00:26:06.090 --> 00:26:24.790
if a conditional expectation of the random
variable at n+1 there is X n+1 having the
00:26:24.790 --> 00:26:34.370
information till time n is X n.
00:26:34.370 --> 00:26:43.910
So how it is related to gambling possibly?
It says that a stochastic process is a Martingale
00:26:43.910 --> 00:26:53.500
if your expected payoff at the n+1 th move
is same as whatever payoff you have received
00:26:53.500 --> 00:26:59.960
at the nth time provided you have only information
up to nth time which is that fact. You will
00:26:59.960 --> 00:27:05.400
not know what will happen at n+1 it is uncertain.
Up to nth time you have information, you know
00:27:05.400 --> 00:27:14.070
that this is X n is what I have got is a random
variable or the random process which was revealed.
00:27:14.070 --> 00:27:21.360
When I had done the gambling at nth instant
but when I am going to do at the n+1th instant
00:27:21.360 --> 00:27:28.830
conditioning upon the fact that this is known
then this is my answer, when I cannot expect
00:27:28.830 --> 00:27:34.120
to improve. Once I improve then I call it
something like a super Martingale if I do
00:27:34.120 --> 00:27:36.850
bad then I call it like a sub Martingale.
00:27:36.850 --> 00:27:46.360
So this is what one should expect that I can
only expect I will do as much as I did last
00:27:46.360 --> 00:27:58.280
time. My payoff would be as much as I had
last time, nothing else. Of course you can
00:27:58.280 --> 00:28:40.970
ask that, this will go on basically, n-2 F
n+1 this will again become X n+1. So this
00:28:40.970 --> 00:28:48.130
is what is known to me. Now I will put this
fact here. Let me see what I get out of this.
00:28:48.130 --> 00:28:51.080
Some little game with conditional expectation.
00:28:51.080 --> 00:29:01.900
So this itself is again this. So if a scenario
has revealed at the n+1 th time then what
00:29:01.900 --> 00:29:09.470
I expect to get for a given scenario is same
as what I would get at the previous time.
00:29:09.470 --> 00:29:15.540
That is the meaning of, I cannot expect something
more. I have to remain status quo. That is
00:29:15.540 --> 00:29:41.140
what I can expect. See if I am putting this
fact here
00:29:41.140 --> 00:29:47.320
then by using the tower law of conditional
expectation because F n+1 is the largest Sigma-algebra
00:29:47.320 --> 00:29:55.560
than F n. I can write this is nothing but
the conditional expectation of X n+2 conditioned
00:29:55.560 --> 00:29:56.990
on F n.
00:29:56.990 --> 00:30:08.020
So it does not matter whether it is now n+2
or n+5 or n+ 6 you can so this is an application
00:30:08.020 --> 00:30:28.070
of the tower law. So it does not matter what
is your X n, after X n however large n you
00:30:28.070 --> 00:30:35.990
put here put any n does not matter n is n+5
or n+100 or n+10. If you just have information
00:30:35.990 --> 00:30:46.980
up to n level move whatever X n you put n
is any number bigger than n, conditional expectation
00:30:46.980 --> 00:30:52.870
of X m conditioned on the fact that I know
only things up to n is nothing but X n.
00:30:52.870 --> 00:30:58.290
That is the meaning of the Martingale and
this idea this fact that does not matter whatever
00:30:58.290 --> 00:31:03.730
you put here. If you just know up to F n you
know you can only expect what you have got
00:31:03.730 --> 00:31:10.070
at the nth level and this fact this beautiful
fact has some beautiful properties and so
00:31:10.070 --> 00:31:20.390
this idea actually allows you to also look
into the continuous case. If you have a continuous
00:31:20.390 --> 00:31:24.600
Martingale you just have to write this.
00:31:24.600 --> 00:31:38.870
So a continuous stochastic process X t given
the filtration F s would remain to be X s
00:31:38.870 --> 00:31:50.280
whenever s is less than equal to t. So this
is the meaning of a, this is just the definition
00:31:50.280 --> 00:32:08.850
of a continuous Martingale okay.
00:32:08.850 --> 00:32:18.100
Now I will just tell you one property is that
it does not matter whatever be your position.
00:32:18.100 --> 00:32:23.290
If you take the expected value just calculate
this expectation of any one of the random
00:32:23.290 --> 00:32:28.460
variables that is the expectation for each
of the random variables. So you have the stochastic
00:32:28.460 --> 00:32:34.740
process X 1, X 2, X n like this so X 1 X 2
dot dot upto to capital X N, X of capital
00:32:34.740 --> 00:32:40.630
N then if you take each of them as separate
random variables and take their expectation
00:32:40.630 --> 00:32:48.110
provided that this sequence is a Martingale
then every random variable will have the same
00:32:48.110 --> 00:32:49.110
expectation.
00:32:49.110 --> 00:32:55.110
So this calculation can be done very simply,
I will just do it on the top and end our talk
00:32:55.110 --> 00:33:02.670
today and tomorrow we will start some lovely
thing called Brownian motion and that is the
00:33:02.670 --> 00:33:06.760
hardcore stuff because Brownian motion is
to be known because that will allow us to
00:33:06.760 --> 00:33:17.540
model the behavior of stock prices in a financial
market in a stock exchange possibly.
00:33:17.540 --> 00:33:31.341
So here we just do it in a very simple way.
So take expectation of X n so this okay I
00:33:31.341 --> 00:33:35.440
am just writing maybe I should write in a
much more different way but okay I am just
00:33:35.440 --> 00:33:55.890
writing in a standard symbolic way. So this
is nothing but integral but by the very definition
00:33:55.890 --> 00:34:01.890
of conditional expecta tion when you do it
for the general case general Sigma-algebra
00:34:01.890 --> 00:34:06.910
F n not the one generated by a partition you
know you have to use this partial averaging
00:34:06.910 --> 00:34:14.109
idea that is essentially you define it in
that way. This is nothing this one is nothing
00:34:14.109 --> 00:34:22.730
but X n+1 d P. This is the definition of conditional
expectation for the general Sigma-algebra.
00:34:22.730 --> 00:34:31.579
This is called the partial averaging and this
is nothing but expectation of X n+1. So does
00:34:31.579 --> 00:34:39.269
not matter. So which means if X 0 is equal
to X 1 is equal to E of X 2 so what you finally
00:34:39.269 --> 00:34:50.069
get is E of X 0 putting N equal to 0 that
is equal to E of X 1 that is equal to E of
00:34:50.069 --> 00:35:03.700
X 2. So with this lovely property of the Martingale
we will stop our discussion here and tomorrow
00:35:03.700 --> 00:35:11.069
we are going into this wonderful stuff called
Brownian motion and we will take our discussion
00:35:11.069 --> 00:35:19.079
off from there. Thank you.