WEBVTT
Kind: captions
Language: en
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So we are gradually getting deeper and deeper
into finance and looking at the type of people
00:00:21.609 --> 00:00:30.750
who have really enrolled for the course getting
some 1 or 2 mails from them I have decided
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to if you look at the course format I have
decided to change it a little bit and take
00:00:35.390 --> 00:00:39.620
you slightly more into finance so that you
get the finance flavor.
00:00:39.620 --> 00:00:48.790
So here we are going to start getting more
deeper into the tools for mathematical finance
00:00:48.790 --> 00:00:57.000
and then we will end our show with 2 last
lectures which would be on the Black Scholes
00:00:57.000 --> 00:01:01.960
formula for pricing a European call option.
Of course, then we will try to describe what
00:01:01.960 --> 00:01:08.549
is a call option and etc., etc. and what are
the things, how to go ahead with the things.
00:01:08.549 --> 00:01:31.229
So we are now going to talk about a Ito integral
for a general integral.
00:01:31.229 --> 00:01:47.310
So this is no longer a simple process right.
It is a it is essentially a general one. If
00:01:47.310 --> 00:01:57.200
you look back and go back to the sections
on expectation, then you will see we have
00:01:57.200 --> 00:02:02.929
tried to define an integral in the following
way that if you want to define an integral
00:02:02.929 --> 00:02:14.690
of a random variable defined over a probability
space then this has been defined as a limit
00:02:14.690 --> 00:02:22.300
okay of these integrals.
00:02:22.300 --> 00:02:40.290
Limit Y is simple and Y is less than equal
to x. So it is a limit but sorry it is essentially
00:02:40.290 --> 00:02:45.500
not the limit but supremo because you are
taking Y less than equal to x so the integral
00:02:45.500 --> 00:02:51.300
of Y is less than integral of x and then you
are taking the supremo but it is a limit in
00:02:51.300 --> 00:02:56.730
some sense you can also put it in a limit
form. So this is what you have learnt. So
00:02:56.730 --> 00:03:00.230
the same idea should come here.
00:03:00.230 --> 00:03:08.190
So as you keep on changing the partition of
the interval 0 to t for each partition you
00:03:08.190 --> 00:03:16.020
can define 1 particular simple process and
each as you make the partition smaller and
00:03:16.020 --> 00:03:28.140
smaller the simple process will actually go
closer and closer to the suppose this is your
00:03:28.140 --> 00:03:41.960
actual process up to T. Now you have divided
into say this t0 which is 0 t1, t2, t3, t4,
00:03:41.960 --> 00:03:46.909
with t5 and so on and this is t6.
00:03:46.909 --> 00:03:57.780
So what you do is you define 1 value here
maybe then you define another value here in
00:03:57.780 --> 00:04:22.790
this way sorry from t2 maybe like this, from
t3 might be like this t3 to t4 something like
00:04:22.790 --> 00:04:33.270
this, from t4 it might be just like this and
from t5 it might be just like this. So this
00:04:33.270 --> 00:04:39.780
is very crude approximations very bad approximations
but as you make the thing smaller and smaller
00:04:39.780 --> 00:04:45.800
and smaller these partitions your approximations
would start becoming better and better and
00:04:45.800 --> 00:04:46.800
better.
00:04:46.800 --> 00:04:56.340
So what do I mean by that that a simple process
delta n t so I will have a sequence of simple
00:04:56.340 --> 00:05:09.780
process delta n t each process is simple.
So if I take this sequence of simple processes
00:05:09.780 --> 00:05:20.590
as n goes to infinity
we expect that this simple process simple
00:05:20.590 --> 00:05:29.180
process should somehow be same. Of course
you have to define what is the meaning of
00:05:29.180 --> 00:05:33.110
this sameness, same as this continuously varying
process delta t.
00:05:33.110 --> 00:05:40.350
So these are discrete processes whose limiting
form is this continuous process delta t. Continuous
00:05:40.350 --> 00:05:45.130
not a continuous process this continuous process
here but okay continuous more continuous than
00:05:45.130 --> 00:05:51.270
the things that we are talking about the parts
are continuous as they are continuous parts.
00:05:51.270 --> 00:05:55.550
So there is nothing like every step is discrete.
So it may not be just a constant function
00:05:55.550 --> 00:05:58.280
on the interval, it could be just varying.
00:05:58.280 --> 00:06:04.440
So we expect that this is somehow what we
will get but what is the meaning of this sameness
00:06:04.440 --> 00:06:10.530
how do we talk about this sameness. In probability
theory, the sameness is spoken in a slightly
00:06:10.530 --> 00:06:20.590
different way. It tells you I expect that
the distance between these can be made smaller
00:06:20.590 --> 00:06:34.110
and smaller and smaller. So it tells me, now
each of these
00:06:34.110 --> 00:06:39.820
so what we expect is the following. I will
tell you what it means.
00:06:39.820 --> 00:07:03.979
So Shreve tells us that if by these functions
simple functions coming to delta t it means
00:07:03.979 --> 00:07:13.449
that it what it means that over all the sample
paths if you average the square of the distance.
00:07:13.449 --> 00:07:33.479
So this is nothing but the L2 norm. If you
look at this integral. So in the terms of
00:07:33.479 --> 00:07:42.960
the function this is nothing but delta n - delta
whole square and those who know about the
00:07:42.960 --> 00:07:51.050
L2 spaces the space of all square integrable
measurable square integrable functions.
00:07:51.050 --> 00:07:56.910
So once you have and you expect so let us
forget this thing and let us not bother because
00:07:56.910 --> 00:08:02.199
people might be not so comfortable with it.
Let us just look into this. Shreve tells us
00:08:02.199 --> 00:08:09.199
that when I am expecting the fact that I expect
that when n grows this simple processes must
00:08:09.199 --> 00:08:16.970
somehow manage to imitate or will be almost
same as the given process.
00:08:16.970 --> 00:08:22.889
By this statement we are actually meaning
this statement, we are meaning that if we
00:08:22.889 --> 00:08:41.240
take over all the paths we take the distance
between these 2 by the square of the distance
00:08:41.240 --> 00:08:51.520
then the expected value of the distance between
these two points should be 0 because remember
00:08:51.520 --> 00:08:57.690
in a random setting we cannot be telling that
it is exactly equal to 0, this will be exactly
00:08:57.690 --> 00:09:03.610
0. We have to talk in terms of expectation
because we do not know the outcome. So we
00:09:03.610 --> 00:09:06.440
have to only speak in terms of expectation.
00:09:06.440 --> 00:09:16.530
So the expected distance so this is this distance
so as n grows the expected distance should
00:09:16.530 --> 00:09:22.160
be going to 0 because the expected distance
itself is a random variable so the sorry the
00:09:22.160 --> 00:09:26.130
distance itself is a random variable, so the
expected distance between these two has to
00:09:26.130 --> 00:09:33.020
go to 0 that is the idea. So by this we actually
mean this.
00:09:33.020 --> 00:09:52.720
So once we mean this the definition of a standard
integral standard Ito integral from any point
00:09:52.720 --> 00:10:17.600
so would be limit. So it can happen when you
take any t between 0 to t this will happen.
00:10:17.600 --> 00:10:28.990
This is the way I define. Now why such a limit
would exist. Remember that this is a stochastic
00:10:28.990 --> 00:10:38.130
process. I am taking a limit of another stochastic
process. So why should such a limit exist
00:10:38.130 --> 00:10:41.570
right. For every given t why should such a
limit exist.
00:10:41.570 --> 00:10:48.100
That can be answered by using the Ito isometric
theorem and showing that this actually forms
00:10:48.100 --> 00:10:53.270
a Cauchy sequence but we are not going to
get into that sort of argument at all though
00:10:53.270 --> 00:10:59.500
I assume that you must be understanding this
what Cauchy sequence and those who do not
00:10:59.500 --> 00:11:09.900
just forget it for the time being. Just assume
that okay I can calculate it in some procedure.
00:11:09.900 --> 00:11:17.350
Now what sort of properties would such an
integral satisfy. You will be amused that
00:11:17.350 --> 00:11:23.959
all the properties that we learnt about integrating
the simple process would be applicable here.
00:11:23.959 --> 00:11:40.240
For example that
I t is continuous as a function of t for a
00:11:40.240 --> 00:12:08.120
given sample path t for a given scenario omega
t for any given scenario omega. Number 2.
00:12:08.120 --> 00:12:18.769
I t is F t measurable that an I t is a stochastic
process adapted to the filtration F t which
00:12:18.769 --> 00:12:23.279
is the filtration associated with the Brownian
motion.
00:12:23.279 --> 00:12:27.790
From now on whatever filtration we are talking
about the filtration would be associated with
00:12:27.790 --> 00:12:54.750
the Brownian motion. So I t is F t measurable.
The third property is linearity. So if we
00:12:54.750 --> 00:13:07.421
take any 2 processes gamma and u, sorry delta
and gamma it can be proved that this is same
00:13:07.421 --> 00:13:21.170
as we are not going to prove these things
just to announce you that these are the important
00:13:21.170 --> 00:13:43.579
properties. Another fourth most important
property is that I t is a Martingale.
00:13:43.579 --> 00:14:00.060
Then of course we should talk about the Ito
isometry property. So Ito isometry you know
00:14:00.060 --> 00:14:04.550
what is Ito isometry, it is exactly the same.
You basically copy what you know for that
00:14:04.550 --> 00:14:28.639
to this part. The Ito isometry says a second
movement of this
00:14:28.639 --> 00:14:36.540
and the last property which is one of the
most seminal property so I separate it, the
00:14:36.540 --> 00:14:40.451
quadratic variation of the Ito integral I
t.
00:14:40.451 --> 00:14:47.300
Remember Ito integral I t itself I t the Ito
integral again I this is the point I want
00:14:47.300 --> 00:15:00.620
to state again and again the Ito integral
the Ito integral
00:15:00.620 --> 00:15:21.610
is a stochastic process. So the major property
that I want to state is that a quadratic variation
00:15:21.610 --> 00:15:29.660
accumulated up to time t is t. That is independent
of the path quite a Brownian motion type behaviour
00:15:29.660 --> 00:15:31.920
sorry not t ,I am making a mistake here.
00:15:31.920 --> 00:15:40.990
This is not Brownian motion type behaviour
sorry it is the same as same as the isometry.
00:15:40.990 --> 00:15:55.670
So here you see
so it gives you the standard. This is the
00:15:55.670 --> 00:16:03.300
normal integration in terms of the variable
u. So this is a stochastic process. For every
00:16:03.300 --> 00:16:11.850
omega, you take you will get something. So
as omega changes this will change sorry I
00:16:11.850 --> 00:16:18.160
made a mistake in my exit, it is path dependent
actually.
00:16:18.160 --> 00:16:25.800
So this is itself this quadratic variation
itself is a stochastic process just writing
00:16:25.800 --> 00:16:33.829
in short and this is path dependent of course
not path independent I made a mistake sorry.
00:16:33.829 --> 00:16:39.569
So these are the important properties that
you learn about the Ito integral. So these
00:16:39.569 --> 00:16:44.079
are the properties that you have to keep in
mind. Of course, because of the time we cannot
00:16:44.079 --> 00:16:49.230
be proving each and every step. Now what we
will do we will show you a simple example
00:16:49.230 --> 00:16:56.720
of how to compute the Ito integral and we
will make a difference between computing the
00:16:56.720 --> 00:17:00.639
stochastic integral and the standard form
of computing this integral.
00:17:00.639 --> 00:17:09.770
We will just see. So the next part remaining
part of the class would actually consist of
00:17:09.770 --> 00:17:17.540
computing this integral. Please understand
quadratic variation is a very very fundamental
00:17:17.540 --> 00:17:37.550
thing here. So our goal would be to compute.
00:17:37.550 --> 00:17:46.330
So delta omega t we have replaced by W t.
So what would be this? That is basically 0
00:17:46.330 --> 00:17:53.550
to capital T x d x that will be x square by
2 that will be t square by 2 would be the
00:17:53.550 --> 00:18:00.460
answer if I just take it as a standard calculus
integral but you will soon see that t square
00:18:00.460 --> 00:18:11.270
by 2 is not the answer right. We are just
putting x here and dx will be x square by
00:18:11.270 --> 00:18:16.550
2 would be the answer. x square by 2, 0 to
t so t square by 2.
00:18:16.550 --> 00:18:22.050
But here you will see is the quadratic variation
which will get you an additional term with
00:18:22.050 --> 00:18:27.640
a term of that type and that makes stochastic
integrals very different from the standard
00:18:27.640 --> 00:18:36.470
integrals of calculus. So here we do it start
doing it step by step. So we will construct
00:18:36.470 --> 00:18:47.830
for
this particular we will make a construction
00:18:47.830 --> 00:19:01.660
of a sequence of simple processes. Now we
will choose n to be a very large integer so
00:19:01.660 --> 00:19:07.690
n is an integer so we will keep on increasing
n, n is an integer.
00:19:07.690 --> 00:19:19.890
So take an n and have for every step so you
divide every interval length is of the length
00:19:19.890 --> 00:19:35.970
T by n, 0, T by n, 2T by n, and so and so
forth till 3T by n that is all that is all.
00:19:35.970 --> 00:19:44.420
Now as n becomes larger and larger these intervals
become smaller and smaller. So we will now
00:19:44.420 --> 00:19:51.230
define that for every interval how would we
define the delta n t because when interval
00:19:51.230 --> 00:19:55.900
these intervals become smaller and smaller
and the length of this interval that this
00:19:55.900 --> 00:19:59.620
will be larger these lengths would be smaller
and smaller and smaller.
00:19:59.620 --> 00:20:11.350
There will be more and more intervals of those
given lengths. So we will put W 0, it will
00:20:11.350 --> 00:20:29.700
be same = 0, so it will be 0 if ,
00:20:29.700 --> 00:21:13.670
W T by n it is from here W T by n so W let
me just put this is W n-1 T by n. So, you
00:21:13.670 --> 00:21:36.540
are just taking these values itself. So that
is that is what it is, sorry that is T here.
00:21:36.540 --> 00:21:46.290
So that is how I have constructed it. Now
the next question would be to know whether
00:21:46.290 --> 00:22:30.120
such a construction can prove this thing sorry
this will be W t that would become 0.
00:22:30.120 --> 00:22:42.882
So what is the expected value of this? That
is an important question. So you can understand
00:22:42.882 --> 00:22:49.400
if you look at the picture very well I would
advise you to draw the picture yourself then
00:22:49.400 --> 00:23:00.030
that is the way you can learn. If you draw
the picture is a Brownian motion and here
00:23:00.030 --> 00:23:06.560
you have kept say 0 and then you have taken
the t1 value is here you have kept it here
00:23:06.560 --> 00:23:19.300
till say t2 and t2 value say is here started
like this.
00:23:19.300 --> 00:23:28.540
So in that way you will see that as n will
become larger and larger these straight lines
00:23:28.540 --> 00:23:35.990
will become smaller and smaller and smaller
become smaller and smaller and smaller and
00:23:35.990 --> 00:23:42.490
so you will have many of these straight lines
as n becomes large and large there will be
00:23:42.490 --> 00:23:48.110
infinitely many straight lines which would
very hardly vary from the actual value that
00:23:48.110 --> 00:23:58.760
you will see. So this distance would any way
become smaller and smaller.
00:23:58.760 --> 00:24:13.770
So if you take their difference for a given
sample path and integrate over them then basically
00:24:13.770 --> 00:24:19.990
you are calculating of the distance between
the areas under the curves for a given sample
00:24:19.990 --> 00:24:25.340
path square of the distance basically not
really area but square of the distance then
00:24:25.340 --> 00:24:34.110
this would turn out to be 0. Of course you
can make a very rigorous calculation right
00:24:34.110 --> 00:24:40.700
of actually putting in the values breaking
up 0 to t into these intervals and then actually
00:24:40.700 --> 00:24:43.430
looking into the difference W0 - W t.
00:24:43.430 --> 00:24:51.150
When t is lying between this so the square
of the difference you can write them down
00:24:51.150 --> 00:25:01.970
and then you can separately come and do the
job so doing this will take a little bit of
00:25:01.970 --> 00:25:16.580
time. It looks slightly obvious that you can
easily do this but this thing actually getting
00:25:16.580 --> 00:25:21.750
this thing done of course one might think
that how I should be able to change with the
00:25:21.750 --> 00:25:23.740
expectation inside the integral.
00:25:23.740 --> 00:25:30.870
Yeah, you can do under certain circumstances
you can switch expectations and integrals.
00:25:30.870 --> 00:25:35.050
So if you can switch the expectation integral
then the things would be much more simpler
00:25:35.050 --> 00:25:41.610
then you would know that okay their difference
would be 0 their mean would be 0 etc., etc.
00:25:41.610 --> 00:25:48.300
and sorry the variance would be not the mean
that will be the variance so variance would
00:25:48.300 --> 00:25:55.540
be basically t by n and so and so forth.
00:25:55.540 --> 00:26:01.930
So you would add those quantities up right
so you would add those quantities up and then
00:26:01.930 --> 00:26:07.000
you integrate with n becoming larger and larger
so the area would actually keep on falling.
00:26:07.000 --> 00:26:10.720
So that is 1 way of thinking about it but
let us not think about it at this moment.
00:26:10.720 --> 00:26:18.140
Let us assume that we can show this that this
will actually work. So this is a very good
00:26:18.140 --> 00:26:24.180
approximation which it is if you look at it
pictorially which it is.
00:26:24.180 --> 00:26:29.300
You see in actual finance you really have
do not have to bother too much about these
00:26:29.300 --> 00:26:34.450
sorts of approximation nobody is going to
ask you to do such calculations but okay if
00:26:34.450 --> 00:26:43.130
we find time we will really put up these sorts
of things up in the website.
00:26:43.130 --> 00:26:57.830
So now what we have is the following
00:26:57.830 --> 00:27:14.460
that is my definition. So how do I define
this delta you already know. So, it depends
00:27:14.460 --> 00:27:23.580
on
so it is from j equal to n so it is just using
00:27:23.580 --> 00:27:37.280
the values there. At every step you have to
calculate Wj T by n and j it is 1 it is j
00:27:37.280 --> 00:27:45.770
is 0 it is 0, j it is 1 it is W T by n, j
is basically you have to do this because these
00:27:45.770 --> 00:27:59.251
are the function points where these are the
function values. You can easily understand
00:27:59.251 --> 00:28:10.510
these things.
00:28:10.510 --> 00:28:20.320
So what have I done. So when j is 0 it is
0 done finished. When j is 1 it is W T by
00:28:20.320 --> 00:28:29.250
n into W 2t by n - W t by n of course because
over that particular interval it will take
00:28:29.250 --> 00:28:37.620
the value j T by n. Over this interval it
will take the value j t j W this will take
00:28:37.620 --> 00:28:43.730
the value W T by n. So that is exactly the
interval that we are considering. So and so
00:28:43.730 --> 00:28:48.220
forth. So this is the sum.
00:28:48.220 --> 00:28:52.830
So essentially now we have to compute this
sum and then take the limit. So our job now
00:28:52.830 --> 00:29:22.380
is to compute this sum. Now because there
is so much of clumsy things we will consider
00:29:22.380 --> 00:29:34.230
some shorthands. We will write W j shorthand
let us do shorthand some shorthand notations.
00:29:34.230 --> 00:29:49.620
In the shorthand notations let us put W j
to stand for W j T by n so which means W j+1
00:29:49.620 --> 00:29:56.150
would stand for W j +1 T by n.
00:29:56.150 --> 00:30:05.280
So with this shorthand notation we will start
the computation. Now let me do the calculation.
00:30:05.280 --> 00:30:09.910
We have to check up this but let me start
by doing the following calculation. You will
00:30:09.910 --> 00:30:29.350
see why I am doing this. I will just go ahead
and just tell you a bit more about this. Suddenly
00:30:29.350 --> 00:30:34.790
I remembered it is a good way to talk about
it. You see when n is becoming large see the
00:30:34.790 --> 00:30:40.640
first part is very simple because when you
put any t here it does not matter.
00:30:40.640 --> 00:30:46.090
Here W it will be 0 and when n is becoming
very very large because W T is continuous
00:30:46.090 --> 00:30:52.030
and when n is becoming very very large this
thing becomes very small this interval. So
00:30:52.030 --> 00:30:58.780
there is hardly a difference between t and
delta and t these values, very small difference
00:30:58.780 --> 00:31:04.820
and so this square values would actually come
down further and that is why their values
00:31:04.820 --> 00:31:12.690
would basically come down and the expectation
would go towards 0.
00:31:12.690 --> 00:31:18.700
So I want to calculate this. You will see
why we are calculating this. This would allow
00:31:18.700 --> 00:32:01.420
us to get a fairly good this will allow us
to calculate this. So it is just an opening
00:32:01.420 --> 00:32:14.010
of the brackets nothing else. Once I do open
the brackets because W 0 is 0 then so I just
00:32:14.010 --> 00:32:19.280
very important note that W 0 is always 0.
So here because I am putting when I am putting
00:32:19.280 --> 00:32:25.510
j = 0, I get 1 and I am putting j = n - 1
I am getting n.
00:32:25.510 --> 00:32:34.060
So I can as well as write these expression
as this particular expression as k = 1 to
00:32:34.060 --> 00:32:53.900
n, W k square so - j = 0 to n-1 W j W j+1
plus half of summation j = 0 to n-1 W j square.
00:32:53.900 --> 00:33:04.010
So now I will break this up into 2 parts.
I will break this up into W half W n square
00:33:04.010 --> 00:33:12.620
plus summation. Now I will use the fact that
W 0 is 0. So what I will have? If I break
00:33:12.620 --> 00:33:19.260
it up I will have summation k = 1 to n-1 W
k square but noting the fact that W 0 is 0,
00:33:19.260 --> 00:33:28.050
I can add W 0 square to it and write here
this break it as k = 0 to n-1, W k square.
00:33:28.050 --> 00:33:50.090
It does not matter because W k is anyway 0.
00:33:50.090 --> 00:33:56.340
So here is what we have got. But now we will
go and transfer our calculation to this part.
00:33:56.340 --> 00:34:04.050
So we will do our calculation in this part.
So I will have half of summation j = 0 to
00:34:04.050 --> 00:34:13.940
n-1 W j+1 - W j square and again I can begin
up from this part. So what I will do I will
00:34:13.940 --> 00:34:19.790
add sorry there is a half here. I will add
this half and this half because these j’s
00:34:19.790 --> 00:34:21.780
and k’s are dummy indices.
00:34:21.780 --> 00:34:33.270
So I will have half W n square
plus j = 0 so half plus half is added so it
00:34:33.270 --> 00:34:47.560
is 1. n-1 W j square - summation j = 0 to
n-1, W j W j plus 1.
00:34:47.560 --> 00:35:06.000
So basically I am getting half W n square
plus summation j = 0 to n-1 W j into W j - W
00:35:06.000 --> 00:35:13.020
j+1. You see I have almost got this expression
but with a minus sign. So what I will do is
00:35:13.020 --> 00:35:20.000
the following. But what is this? This is a
quadratic variation term. So I will take this
00:35:20.000 --> 00:35:26.390
part to this side and bring this thing to
this side. So my next job is the following.
00:35:26.390 --> 00:35:52.070
So I will now write summation j = 0 to n-1,
W j W j+1 - W j = half W n square - half summation
00:35:52.070 --> 00:36:02.710
j = 0 to n-1, W j+1 - W j square. This is
actually the form of the quadratic variation
00:36:02.710 --> 00:36:09.330
of the Brownian motion you see so quadratic
variation has come.
00:36:09.330 --> 00:36:22.780
So basically, now if I take the limit on both
sides what is this? W n square W n square.
00:36:22.780 --> 00:36:37.230
So what will I get? W n t. So if I put n here
W j W n is W t. So here it will be nothing
00:36:37.230 --> 00:36:45.610
but half W t square. So if I take as n tends
to infinity this limit is nothing but the
00:36:45.610 --> 00:37:03.860
integral. So I will get 0 to t W t d W t = half
W t square - this limit, assuming that all
00:37:03.860 --> 00:37:08.580
the limits that I am taking here I am assuming
that okay there are in almost surely sense.
00:37:08.580 --> 00:37:13.450
You can even take them in the sense of limits
in the sense of probability convergence in
00:37:13.450 --> 00:37:16.480
P that will also do.
00:37:16.480 --> 00:37:25.760
So I am writing so this equality is a convergence
in probability. Essentially this is because
00:37:25.760 --> 00:37:32.850
we can actually show the quadratic variation
goes in or can be shown in almost surely sense.
00:37:32.850 --> 00:37:37.530
So these are actually in almost this equality
is always in almost surely sense because this
00:37:37.530 --> 00:37:44.200
is a stochastic process random variable at
time t so this equality holds only everywhere
00:37:44.200 --> 00:37:45.460
except a set of measured 0.
00:37:45.460 --> 00:37:52.230
So this is very very important to understand
when a set of a null event if you throw away
00:37:52.230 --> 00:37:58.530
null event it holds everywhere. So what do
you get? So you get 0 sorry this is half here.
00:37:58.530 --> 00:38:15.430
So what you get 0 to t W t d t sorry W t d
W t = half W square t - half of t. You see
00:38:15.430 --> 00:38:24.050
this half of t is an additional term because
if I take a standard integral 0 to t x square
00:38:24.050 --> 00:38:30.010
dx sorry x dx. This is nothing but t square
by 2.
00:38:30.010 --> 00:38:40.410
This is nothing but t square by 2. Or if you
take any integral as 0 to t, g t ,d g t. d
00:38:40.410 --> 00:38:51.490
g t is nothing but g dash t, g t, 0 to t ,g
t, g dash t d t because this is the differential
00:38:51.490 --> 00:39:04.540
of t d t and this will simply give you half
g square t. So you see this is something very
00:39:04.540 --> 00:39:12.110
different and that is the crux of stochastic
calculus that stochastic calculus is a quadratic
00:39:12.110 --> 00:39:15.890
variation actually makes the difference the
quadratic variation.
00:39:15.890 --> 00:39:19.830
See essentially, we were trying to compute
the quadratic variation so these are some
00:39:19.830 --> 00:39:25.330
sort of a trick. The quadratic variation cannot
be forgotten when Brownian motion is in the
00:39:25.330 --> 00:39:33.540
game. So we stop here and we will talk about
Ito’s calculus in the next we will have
00:39:33.540 --> 00:39:38.270
a Taylor theorem type things that you learn
in standard calculus. What is the meaning
00:39:38.270 --> 00:39:48.630
of Taylor’s theorem in stochastic calculus?
So we will learn about those things.