So, here we are, on our third week run and
today we are approaching or rather entering the fascinating land of the Ito calculus or
the Ito integral. Ito is the name of a Japanese scientist who introduced this and with that,
he changed many things in physics, his things changed many things in physics and also in
probability and Ito integrals and the calculus associated with it, the Ito calculus, is the
key to doing a lot of computing or solving lot of models in finance. So, this is the holy grail of stochastic calculus
and we are going to do it step by step. We are not going to be in a hurry, we are not
going to push ourselves into too much of rigor. Of course, rigor is a necessary tool of a
mathematician, but looking at the diverse audience that I have, you cannot push rigor
too much, but I will try to get you some understanding and in this whole the remaining part of the
course, the book, I will follow a single book, which is one of my favorite books in mathematical
finance. It is called stochastic calculus for finance. So, this book is written by, Steven Shreve
a very famous name in mathematical finance from Carnegie Mellon University. I have taught
from this book earlier and this is my favorite book in this subject. I would rather like
to show it, this is the book and the interesting part is that, this book now is an Indian edition,
both the volume. This is volume 2 from which I am teaching.
So, volume 1 and 2 there are 2 volumes. Volume 1 talks about discrete time. Here we are talking
about continuous time and both these books are very a lovely read and those who are serious
about mathematical finance should actually procure the copy of these 2 books. This will
be helpful in many-many ways and this book takes you from a very basic to pretty advanced
level. So, now what we are going to do. In this section,
we are trying to make sense of the term, sense of this. Of course, you might ask what is
dWt. We just said in the last class that, this Brownian motion is not differentiable
anywhere. Yes, but assume that this is like writing
the difference. This is the increment of the Brownian motion we and we had been talking
about increments of this Brownian motion. So, we want to so the question is, can we
make sense of this and what is this delta t? Delta t
is a process adapted to the filtration Ft associated with
the Brownian motion. So, just like in standard integration theory, we will start by trying
to look into the meaning of this, when this process is nothing but step functions. At
every t it is some step function and on a particular interval it will hold a particular
value fixed value. So, our job would be now to make you, will
see that, finally when we will compute things, you will see that this will give us very different
answer than standard calculus. It is the quadratic variation of the Brownian motion that would
be the culprit. So, here again, we are going to partition 0 T into intervals. So here we
are partitioning where you have this. Now, we will talk about a simple process or
a simple function basically. We will take this to be a simple process. Simple process
means on each interval tj to tj+1 which includes tj, not tj+1, this is constant. So basically, you are looking at a function
of this form. So here is your 0 to T, here is t1, here is t2, here is t3 say I will just
do it 5, t4, t5 and this is t6. So, what you are expecting that from 0 suppose this, to
here, you have some particular value. Then t1, so here you do not have the value, this
value is not valid here it starts again from here, t1 to t2 that is why you are giving
this round. This is one particular value and t2 to 3 it
could be like this, t3 to t4 it could be like this, t4 to t5 it could be this one, and t5
to t6 it could be like this just. This is a step function basically and this is an example
of a simple process. So, this is one sample path of that simple
process right. So, in this thing, if this is the way you have broken it up the partition,
on each partition we will consider that it is constant. Of course, if you take more smaller
partitions, the same idea will remain fixed. Okay, there will be 2 intervals where you
will have the same value. Now given this partition on which it is constant,
so we will look at only those intervals on which it is constant and that will partition
the interval. We will try to calculate it. Now by delta t let us work like a financial
guy. Let us in delta t is the position I am taking
in an asset at time t that is delta like a stock so delta t is the number of stocks I
am holding at time t of some company for example. So, number of stocks I am
holding at time t. Now let us, now assume that Wt the Brownian motion, just like as
Bachelier thought, describes the price process of the stock. But you know that, this is really
not true because, price process of stocks never take negative values while Brownian
motion can. But for the moment, just for the heck of it,
think that we are in Bachelier’s days, we are not bothering of those things and we are
talking about the fact that W t is the price. So, what I have done so at time t, what is
my gain in trading at time t. So, I t is the gain is the process is actually a stochastic
process, describing the gain by trading at time t. Now consider the inter t, which is lying between
t1 and t0 right. Now I have actually made, I have actually, bought say delta t0 quantity
of objects at time t0, and for which I had paid Wt0 as the price. So, Wt0 is the price
per unit of course. So, my total money I paid, so let me write I t so what is I t in this
case. So, total money I paid, to get this delta
t0 amount of stocks at t0 is W t0 and total money now I take hold on to this stock at
time t at a time k to sell that number of stocks and at a price Wt, because price is
changing at every time. So, by selling those stocks, I make this amount of money. So, what
is my gain? Gain is nothing but the difference of this two. Amount of money I spent and then
amount of money I actually bought by selling, so I will just subtract these two to know
whether I am at profit or loss, gain could be negative also. Now think that what would happen if my t,
is between t2 what would happen to this, what would happen to this scenario right. So, what
would be I t when I am in that particular scenario. In that case, of course t0 is in
this case 0, so we can write delta 0 also here, does not matter. So, when I am writing
this t here, one has to be very careful. I am meaning that, I am ending the trading at
this time. So, what happens when I am telling that t
is between t1 and t2 because which means I have held to the stock Wt0 up to the time
t1. After that, I will change it, so at time t1, I sell my stock which I bought, this amount
of stock at time 0 with the price Wt1. Then I bought, a stock with this money that
I have got, I have bought a stock Wt1 delta t1 at time t1, paying price Wt1 then at t2,
at time capital T which is lying between t1 and t2 I sell this this Wt1 holding, a delta
t1 holding so delta t1 Wt. So, by selling I totally gained this amount and this is the
amount I have spent to actually buy the delta t1 stocks. So, it will be this minus this.
So, which I can also write nicely as I t in this particular case I t = delta 0 W t1 +
delta t1 W t - W t1. Again, if you are slightly not comfortable,
what I did, I know that my maximum time I can hold the stock, I bought at time 0 is
up to time t1, then I have to sell it. That is the period of holding. Then at time t1,
I know what is the stock price I may sell it with this price. Now with that price I
buy, stocks W t1 stock of the amount delta t,1 with the price W t1. So, this amount now
has to be subtracted from this one, because I sold and got some money, from that I spent
some money to buy some new stocks. Again, I sold out stocks at time capital T
and I finish my trading, time small t and I finish my trading. I got this amount of
money. So, this minus this plus this is the amount of money I have now. So, this is a very simple thing. Now this
can be written in a more general way, that if t, so if you have a t lying between t k
and t k+1 then your I t, so this is what does it say. Say at every time point, before the
time, till time k-1, what did he do? He bought something, he hold something in 1 interval,
then sold that thing and bought the new stock by using the money from the same thing that
he had gained. When he sold something whatever money he had
got he spent a part of it to buy something. So, he held at this stock, up to time t j+1
and sold this at this amount, but at time tj, he had at, sorry at time t j he had bought
the same amount by spending the money delta t j into W t j. And then of course, we will
look at the last part here this into this whole thing into Wt minus Wtk. So, this is the general form of writing I
t. if you are not comfortable with, the cannot see it I write it again here for your convenience.
This is very simple profit and loss thing. I trade make money use that money to buy something,
again I sell that and buy some new thing. So, that is the whole thing that is whole
idea that is the way the market actually operates. This I t, so this I t is often written as
I 0 to small t. So, this whole thing is written in a shorthand like this. That is how you
give meaning to this thing. So, this is what we want to write as dWt. I t is a shorthand
writing. I t is not exactly the differential. There is nothing like a differential of Brownian
motion. It is a shorthand writing. Now I t then itself is a stochastic process. So, I t is called the Ito integral of the
simple process delta t. So, you are taking integral of a stochastic process that is very
important and it is not like integrating a random variable. Integrating random variable
is just like integrating any function right that is it. So, here what you create is also a stochastic
process. Your integral itself is not a number but your integral itself is also a stochastic
process. I t is a stochastic process. Son you see there is a huge leap or huge difference
in the thought process. Son when you come to this stochastic wall your life completely
changes. Now what are the key properties of this stochastic
process. The key properties are following. The number 1 property is that, the Ito integral
properties, if we have time we will prove any one of them, the number 1 property is
that, Ito integral is a Martingale with respect to the filtration Ft adapted to the Brownian
motion W. The Ito integral is a Martingale. Ito integral, so when I am taking integral,
do not think it is an integration, it is a number. It is not definite integral, Ito integral
I t means, Ito integral is a stochastic process. Ito integral I t, is a Martingale. Martingale
are essentially processes which possibly are trying to maintain status quo at every time. Number 2, proving this is not so simple. It
needs little bit of work nothing else, it is just the computation is complex. Another
thing is called Ito isometry. We will see, what is the meaning of Ito isometry. Ito isometry
says the following. Expectation of I square t, so expectation of the random variable I
square t square, is equal to expectation of 0 to t delta square u du. Now why it is called isometry, would be clear,
if you only know something about L2 norms. Say look at this quantity, E I square t. So,
if I write here as E I square t this simply means integral right, whatever
is 0 to t, I am sorry, integral over, sample space whatever the sample space is of I square
t dP. So, this because, we are assuming that this
is finite. So, once you assume that this is finite you are assuming that I square t actually
belongs with a capital L2 of, L2 space of, this is only for those who know this stuff,
this function analysis. Those who do not do not bother. Think that Ito divides this formula. Isometry means they are similar distances.
So, distance in the original space between 2 points and distance in the range space between
the 2 functional values are similar, then we call it is isometry. So, this actually
means if you look at it very carefully, it is, if you take I, no sorry no, not I, I t
is in L2 f not I square. So, I t this is a square of the L2 norm. So, this is nothing
but the square of the L2 norm. So, this is I 2 is in infinite dimensional
space in this when I t itself is so the Ito integral itself is an element of L2. So, it
is in the infinite dimensional space is an infinite dimensional object. So, it is so
interesting things are coming in and if you look at this. So, they are telling that this is expectation
of this. So, you are telling that the distance, the norm of this is computed, is nothing but
the expected value of this integral, because this integral, if this the expected value
of this integral means what? This is a normal integral. This is a standard level integration
right. Here, I am not treating delta square as the
function of omega basically, keeping the u fixed, but I am treating it as a function
of u itself. So, for every sample path you can compute this as t changes fix up the sample
path, the scenario omega and you can compute this and then you take expectation over all
the sample path, average over all the sample path. That is exactly same as the L2 norm
of this. That is the idea. Now comes a very-very important property.
So here, all these things are being done for, the Ito integral for the simple process delta
t. We have not spoken about Ito integral for a general case and that we will come tomorrow,
in the next rather in the next class. The very important thing, that we should now understand,
that just like Brownian motion, the Ito integral itself has a quadratic variation and what
is that quadratic variation. So, Ito integral does not have a 0-quadratic
variation. Ito integral itself has a quadratic variation. The Ito integral states, the quadratic
variation of the Ito interval is being given by this simple thing it is amazing the answer.
So, the Ito isometry says the following, that the L2 norm of I the integral, is nothing
but the expectation of the quadratic variation. See the quadratic variation itself, whether
it is of a Brownian motion or whether of this integral itself is a stochastic process. That
is the thing that you have to take into account. The quadratic variation, but it is very interesting,
the quadratic variation has a stochastic process, but for every t it gives you a number, every
t it is a fixed number, if for a given sample path, for every omega, it will calculate a
number very nicely. So, and for every omega whatever be your omega whatever
be your path for a given t, therefore whatever be the omega, this will be the answer. So, it is path independent essentially. So,
the quadratic variation itself is a stochastic process. So, what we have now learnt, is completely
for the case, where delta t is that simple function, that is all. Delta t is this so,
I know that delta t given the t delta t varies like this. So, given any omega if it is in
the time interval, this delta t will have this values that is all. It does not depend
on what omega and so delta t is the simplest simplest function. Delta t sorry, I made a mistake. So, given
an omega delta t will take this this this this values in this interval. If you change
the omega, it will take some other values in this fixed intervals. So, the intervals
are fixed and in those intervals, it is taking constant value but it will depend on the omega.
If you change the omega you will have different values and that omega will change the value
of I t. So, that making I t a stochastic process. Similarly, this quadratic variation itself
is a stochastic process. So, this is a very-very important understanding
that we are essentially generating not numbers, we are at every step generating a stochastic
process. So, this is something of very important thing that we have to keep in mind. So, when
we are going to talk about general integral, this is some adapted process. Delta t is an
adapted process, not a simple process like this, just not say function type thing then,
we proceed as we do in integration theory. We write the, we approximate the given function
delta t, in as a, we approximate it by a family of simple processes, simple functions, and
then you take the integral over those simple functions and then take the limit. That is the idea, we are going to apply and
then we will see for one particular calculation what will happen and there is a so, we had
learnt in the last class that a quadratic variation
of course, we should ask what is the, what is when we take the limits of those Q pi what
was what sort of limit it was. We proved that what the limit we get is a
limit in or the convergence that we get that we can always have convergence in probability.
But under slight, and also if we just bother about the fact
the variance is 0 and the expectation is t then we have something called L2 convergence,
which I am not going to bother you with. Or else by slightly modifying the definition,
slightly modifying the requirements, of the behavior of the partition, we can show that
it can be made to be convergent almost surely. So, you just have to remember this. But remember
this is also a process stochastic process. Sometimes, now because we have started using
this d Wt symbol, we will use this shorthand. This actually means quadratic variation. This
is this is the meaning of this. We will very, this is nothing but these two difference right
the difference square basically. So, we will use this shorthand. Please note that this
thing, whole thing, is a shorthand, shorthand to write quadratic variation. But this shorthand
would be very helpful when you do finance. They are extremely useful to use this shorthand.
Thank you and tomorrow we are going to talk about, some more interesting facts doing,
Ito integral for a general integral, and all these properties that you have learnt here,
would be actually translated to that general integral and then we will do an example.
