WEBVTT
00:00:00.100 --> 00:00:02.500
The following content is
provided under a Creative
00:00:02.500 --> 00:00:04.019
Commons license.
00:00:04.019 --> 00:00:06.360
Your support will help
MIT OpenCourseWare
00:00:06.360 --> 00:00:10.730
continue to offer high quality
educational resources for free.
00:00:10.730 --> 00:00:13.340
To make a donation or
view additional materials
00:00:13.340 --> 00:00:17.217
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:17.217 --> 00:00:17.842
at ocw.mit.edu.
00:00:21.122 --> 00:00:22.080
PROFESSOR: Let's begin.
00:00:26.540 --> 00:00:31.190
Today we're going to continue
the discussion on Ito calculus.
00:00:31.190 --> 00:00:33.840
I briefly introduced you
to Ito's lemma last time,
00:00:33.840 --> 00:00:38.510
but let's begin by
reviewing it and stating it
00:00:38.510 --> 00:00:39.910
in a slightly more general form.
00:00:42.570 --> 00:00:45.790
Last time what we did was we
did the quadratic variation
00:00:45.790 --> 00:00:51.830
of Brownian motion,
Brownian process.
00:00:57.140 --> 00:01:01.160
We defined the Brownian
process, Brownian motion,
00:01:01.160 --> 00:01:11.700
and then showed that it has
quadratic variation, which
00:01:11.700 --> 00:01:22.380
can be written in this form--
dB square is equal to dt.
00:01:22.380 --> 00:01:33.350
And then we used that to show
the simple form of Ito's lemma,
00:01:33.350 --> 00:01:38.080
which says that if f is a
function on the Brownian
00:01:38.080 --> 00:01:48.820
motion, then d of f is
equal to f prime of dB_t
00:01:48.820 --> 00:01:56.990
plus f double prime of dt.
00:01:56.990 --> 00:02:03.730
This additional term was a
characteristic of Ito calculus.
00:02:03.730 --> 00:02:06.405
In classical calculus
we only have this term,
00:02:06.405 --> 00:02:08.509
but we have this
additional term.
00:02:08.509 --> 00:02:10.389
And if you remember,
this happened exactly
00:02:10.389 --> 00:02:13.690
because of this
quadratic variation.
00:02:13.690 --> 00:02:17.100
Let's review it, and let's do
it in a slightly more general
00:02:17.100 --> 00:02:18.530
form.
00:02:18.530 --> 00:02:20.630
As you know, we
have a function f
00:02:20.630 --> 00:02:24.690
depending on two
variables, t and x.
00:02:24.690 --> 00:02:30.830
Now we're interested
in-- we want to evaluate
00:02:30.830 --> 00:02:35.956
our information on the
function f(t, B_t).
00:02:38.890 --> 00:02:41.120
The second coordinate,
we're planning
00:02:41.120 --> 00:02:43.966
to put in the
Brownian motion there.
00:02:43.966 --> 00:02:45.590
Then again, let's do
the same analysis.
00:02:45.590 --> 00:02:52.530
Can we describe d of f in terms
of these differentiations?
00:02:52.530 --> 00:02:58.610
To do that, deflect this, let
me start from Taylor expansion.
00:03:04.460 --> 00:03:16.080
f at a point t plus delta
t, x plus delta x by Taylor
00:03:16.080 --> 00:03:31.400
expansion for two variables is
f of t of x plus partial of f
00:03:31.400 --> 00:03:39.068
over partial of t at t
comma x of delta t plus...
00:03:42.980 --> 00:03:44.510
x.
00:03:44.510 --> 00:03:47.651
That's the first-order terms.
00:03:47.651 --> 00:03:49.150
Then we have the
second-order terms.
00:04:27.560 --> 00:04:31.440
Then the third-order
terms, and so on.
00:04:31.440 --> 00:04:34.700
That's just Taylor expansion.
00:04:34.700 --> 00:04:37.030
If you look at it,
we have a function f.
00:04:37.030 --> 00:04:39.370
We want to look at the
difference of f when we change
00:04:39.370 --> 00:04:41.661
the first variable a little
bit and the second variable
00:04:41.661 --> 00:04:42.830
a little bit.
00:04:42.830 --> 00:04:44.984
We start from f of t of x.
00:04:44.984 --> 00:04:47.400
In the first-order terms, you
take the partial derivative,
00:04:47.400 --> 00:04:50.350
so take del f over del
t, and then multiply
00:04:50.350 --> 00:04:52.200
by the t difference.
00:04:52.200 --> 00:04:54.350
Second term, you take
the partial derivative
00:04:54.350 --> 00:04:57.420
with respect to the second
variable-- partial f
00:04:57.420 --> 00:05:02.470
over partial x-- and
then multiply by del x.
00:05:02.470 --> 00:05:05.560
That much is enough
for classical calculus.
00:05:05.560 --> 00:05:08.000
But then, as we
have seen before,
00:05:08.000 --> 00:05:09.750
we ought to look at
the second-order term.
00:05:09.750 --> 00:05:14.030
So let's first write
down what it is.
00:05:14.030 --> 00:05:16.370
That's exactly what happened
in Taylor expansion,
00:05:16.370 --> 00:05:17.140
if you remember.
00:05:17.140 --> 00:05:20.130
If you don't remember,
just believe me.
00:05:20.130 --> 00:05:23.856
This 1 over 2 times, take the
second derivatives, partial.
00:05:26.400 --> 00:05:29.100
Let's write it in
terms of-- yes?
00:05:29.100 --> 00:05:31.806
AUDIENCE: [INAUDIBLE]
00:05:34.464 --> 00:05:36.978
PROFESSOR: Oh,
yeah, you're right.
00:05:36.978 --> 00:05:37.946
Thank you.
00:05:45.700 --> 00:05:46.350
Is it good now?
00:05:50.060 --> 00:05:52.150
Let's write it as
dt, all these deltas.
00:05:56.592 --> 00:05:57.830
I'll just write like that.
00:05:57.830 --> 00:06:00.270
I'll just not
write down t and x.
00:06:00.270 --> 00:06:05.980
And what we have is f plus
del f over del t dt plus del
00:06:05.980 --> 00:06:10.810
f over del x dx plus
the second-order terms.
00:06:37.290 --> 00:06:39.990
The only important terms--
first of all, these terms
00:06:39.990 --> 00:06:42.620
are important.
00:06:42.620 --> 00:06:44.660
But then, if you want
to use x equals B of
00:06:44.660 --> 00:06:49.780
t-- so if you're now
interested in f t comma B of t.
00:06:49.780 --> 00:06:55.970
Or more generally, if you're
interested in f t plus dt,
00:06:55.970 --> 00:07:03.285
f B_t plus d of B_t, then
these terms are important.
00:07:03.285 --> 00:07:07.720
If you subtract f of
t of B_t, what you get
00:07:07.720 --> 00:07:11.460
is these two terms.
00:07:11.460 --> 00:07:16.916
Del f over del t dt
plus del f over del
00:07:16.916 --> 00:07:19.970
x-- I'm just writing
this as a second variable
00:07:19.970 --> 00:07:22.035
differentiation-- at dB_t.
00:07:25.880 --> 00:07:28.550
And then the second-order terms.
00:07:28.550 --> 00:07:32.605
Instead of writing it all down,
dt square is insignificant,
00:07:32.605 --> 00:07:37.580
and dt comma-- dt times
dB_t also is insignificant.
00:07:37.580 --> 00:07:39.910
But the only thing that
matters will be this one.
00:07:39.910 --> 00:07:45.000
This is dB_t square, which
you saw is equal to dt.
00:07:48.920 --> 00:07:52.990
From the second-order term,
we'll have this term surviving.
00:07:52.990 --> 00:08:01.160
1 over 2 partial f over partial
x second derivative, of dt.
00:08:01.160 --> 00:08:04.010
That's it.
00:08:04.010 --> 00:08:05.860
If you rearrange
it, what we get is
00:08:05.860 --> 00:08:18.259
partial f over partial
t plus 1/2 this plus--
00:08:18.259 --> 00:08:19.550
and that's the additional term.
00:08:25.150 --> 00:08:28.620
If you ask me why these
terms are not important
00:08:28.620 --> 00:08:33.150
and this term is important, I
can't really say it rigorously.
00:08:33.150 --> 00:08:36.929
But if you think about dB_t
square equals dt, then d times
00:08:36.929 --> 00:08:39.210
B_t is kind of like
square root of dt.
00:08:39.210 --> 00:08:40.860
It's not a good
notation, but if you
00:08:40.860 --> 00:08:45.870
do that-- these two terms are
significantly smaller than dt
00:08:45.870 --> 00:08:48.030
because you're
taking a power of it.
00:08:48.030 --> 00:08:51.905
dt square becomes a lot
smaller than dt, dt to the 3/2
00:08:51.905 --> 00:08:54.630
is a lot smaller than dt.
00:08:54.630 --> 00:08:59.601
But this one survives because
it's equal to dt here.
00:08:59.601 --> 00:09:01.225
That's just the
high-level description.
00:09:05.530 --> 00:09:08.320
That's a slightly more
sophisticated form
00:09:08.320 --> 00:09:09.470
of Ito's lemma.
00:09:09.470 --> 00:09:12.370
Let me write it down here.
00:09:12.370 --> 00:09:14.513
And let's just fix it now.
00:09:18.441 --> 00:09:48.880
If f of t of B_t-- that's d of
f is equal to-- Any questions?
00:09:58.610 --> 00:10:01.360
Just remember, from the
classical calculus term,
00:10:01.360 --> 00:10:05.385
we're only adding
this one term there.
00:10:05.385 --> 00:10:05.884
Yes?
00:10:05.884 --> 00:10:09.580
AUDIENCE: Why do
we have x there?
00:10:09.580 --> 00:10:15.400
PROFESSOR: Because the second
variable is supposed to be x.
00:10:15.400 --> 00:10:18.316
I don't want to write down
partial derivative with respect
00:10:18.316 --> 00:10:21.240
to a Brownian motion here
because it doesn't look good.
00:10:24.190 --> 00:10:26.490
It just means, take the
partial derivative with respect
00:10:26.490 --> 00:10:28.910
to the second term.
00:10:28.910 --> 00:10:33.340
So just view this as a
function f of t of x,
00:10:33.340 --> 00:10:42.080
evaluate it, and then plug
in x equal B_t in the end,
00:10:42.080 --> 00:10:44.250
because I don't want to
write down partial B_t here.
00:10:51.234 --> 00:10:51.900
Other questions?
00:11:11.810 --> 00:11:27.436
Consider a stochastic process
X of t such that d of X
00:11:27.436 --> 00:11:32.030
is equal to mu times d of t
plus sigma times d of B_t.
00:11:35.360 --> 00:11:38.160
This is almost like
a Brownian motion,
00:11:38.160 --> 00:11:39.720
but you have this
additional term.
00:11:39.720 --> 00:11:41.221
This is called a drift term.
00:11:46.130 --> 00:11:53.245
Basically, this happens if X_t
is equal to mu*t plus sigma
00:11:53.245 --> 00:11:55.710
of B_t.
00:11:55.710 --> 00:11:57.714
Mu and sigma are constants.
00:12:01.387 --> 00:12:02.970
From now on, what
we're going to study
00:12:02.970 --> 00:12:08.390
is stochastic process of
this type, whose difference
00:12:08.390 --> 00:12:12.608
can be written in terms of drift
term and the Brownian motion
00:12:12.608 --> 00:12:13.107
term.
00:12:16.100 --> 00:12:18.165
We want to do a slightly
more general form
00:12:18.165 --> 00:12:21.780
of Ito's lemma, where we
want f of t of X_t here.
00:12:25.502 --> 00:12:27.085
That will be the
main object of study.
00:12:31.593 --> 00:12:34.105
I'll finally state the
strongest Ito's lemma
00:12:34.105 --> 00:12:35.244
that we're going to use.
00:12:44.924 --> 00:12:54.370
f is some smooth function and
X_t is a stochastic process
00:12:54.370 --> 00:12:56.151
like that.
00:12:56.151 --> 00:12:57.095
X_t satisfies...
00:13:06.230 --> 00:13:08.770
where B_t is a Brownian motion.
00:13:08.770 --> 00:13:17.210
Then df of t, X_t
can be expressed
00:13:17.210 --> 00:13:38.510
as-- it's just getting
more and more complicated.
00:13:38.510 --> 00:13:41.580
But it's based on this one
simple principle, really.
00:13:41.580 --> 00:13:45.110
It all happened because
of quadratic variation.
00:13:45.110 --> 00:13:49.800
Now I'll show you why this form
deviates from this form when
00:13:49.800 --> 00:13:58.320
we replace B to an X.
00:13:58.320 --> 00:14:03.400
Remember here all other
terms didn't matter,
00:14:03.400 --> 00:14:08.595
that the only term that mattered
was partial square of f...
00:14:08.595 --> 00:14:11.888
of dx square.
00:14:17.490 --> 00:14:30.990
To prove this, note that df
is partial f over partial t
00:14:30.990 --> 00:14:37.066
dt plus partial f over
partial x d of X_t
00:14:37.066 --> 00:14:42.775
plus 1/2 of d of x squared.
00:14:45.490 --> 00:14:48.970
Just exactly the same, but I've
replaced the dB-- previously,
00:14:48.970 --> 00:14:52.885
what we had dB, I'm
replacing to dX.
00:14:52.885 --> 00:14:58.280
Now what changes is dX_t
can be written like that.
00:14:58.280 --> 00:15:03.580
If you just plug it
in, what you get here
00:15:03.580 --> 00:15:14.010
is partial f over partial
x mu dt plus sigma of dB_t.
00:15:14.010 --> 00:15:19.680
Then what you get here
is 1/2 of partials
00:15:19.680 --> 00:15:23.630
and then mu plus
sigma dB_t square.
00:15:26.620 --> 00:15:31.250
Out of those three terms here
we get mu square dt square
00:15:31.250 --> 00:15:37.590
plus 2 times mu sigma d mu dB
plus sigma square dB square.
00:15:37.590 --> 00:15:40.370
Only this was survives,
just as before.
00:15:40.370 --> 00:15:42.970
These ones disappear.
00:15:42.970 --> 00:15:45.180
And then you just
collect the terms.
00:15:45.180 --> 00:15:48.690
So dt-- there's one dt here.
00:15:48.690 --> 00:15:55.673
There's mu times that here,
and that one will become a dt.
00:15:55.673 --> 00:16:00.308
It's 1/2 of sigma
square partial square...
00:16:00.308 --> 00:16:01.300
of dt.
00:16:01.300 --> 00:16:04.770
And there's only
one dB_t term here.
00:16:04.770 --> 00:16:14.312
Sigma-- I made a mistake, sigma.
00:16:25.080 --> 00:16:27.480
This will be a form that
you'll use the most,
00:16:27.480 --> 00:16:32.710
because you want to evaluate
some stochastic process--
00:16:32.710 --> 00:16:36.150
some function that
depends on time
00:16:36.150 --> 00:16:37.431
and that stochastic process.
00:16:37.431 --> 00:16:39.180
You want to understand
the difference, df.
00:16:42.930 --> 00:16:44.660
The X would have
been written in terms
00:16:44.660 --> 00:16:47.000
of a Brownian motion
and a drift term,
00:16:47.000 --> 00:16:50.090
and then that's the
Ito lemma for you.
00:16:50.090 --> 00:16:51.680
But if you want to
just-- if you just
00:16:51.680 --> 00:16:56.420
see this for the first time,
it just looks too complicated.
00:16:56.420 --> 00:16:59.460
You don't understand where
all the terms are coming from.
00:16:59.460 --> 00:17:01.150
But in reality, what
it's really doing
00:17:01.150 --> 00:17:05.359
is just take this
Taylor expansion.
00:17:05.359 --> 00:17:08.170
Remember these two
classical terms,
00:17:08.170 --> 00:17:11.190
and remember that there's
one more term here.
00:17:11.190 --> 00:17:12.780
You can derive it
if you want to.
00:17:18.990 --> 00:17:21.030
Really try to know
where it all comes from.
00:17:21.030 --> 00:17:28.140
It all started from this one
fact, quadratic variation,
00:17:28.140 --> 00:17:32.940
because that made some of the
second derivative survive,
00:17:32.940 --> 00:17:34.850
and because of those,
you get these kind
00:17:34.850 --> 00:17:35.931
of complicated terms.
00:17:39.180 --> 00:17:39.680
Questions?
00:17:51.165 --> 00:17:53.444
Let's do some examples.
00:17:53.444 --> 00:17:54.110
That's too much.
00:18:02.390 --> 00:18:05.260
Sorry, I'm going to use it
a lot, so let me record it.
00:18:49.671 --> 00:18:54.590
Example number one.
00:18:54.590 --> 00:19:04.270
Let f of x be equal to
x square, and then you
00:19:04.270 --> 00:19:07.160
want to compute d of f at B_t.
00:19:13.590 --> 00:19:16.726
I'll give you three minutes
just to try a practice.
00:19:16.726 --> 00:19:18.030
Did you manage to do this?
00:19:25.610 --> 00:19:26.875
It's a very simple example.
00:19:32.400 --> 00:19:37.740
Assume it's just the
function of two variables,
00:19:37.740 --> 00:19:40.010
but it doesn't depend on t.
00:19:40.010 --> 00:19:44.030
You don't have to do that,
but let me just do that.
00:19:44.030 --> 00:19:45.755
Partial f over partial t is 0.
00:19:49.040 --> 00:19:52.970
Partial f over partial
x is equal to 2x,
00:19:52.970 --> 00:20:01.580
and the second derivative
equal to 2 at t, x.
00:20:01.580 --> 00:20:08.010
Now we just plug in
t comma B_t, and what
00:20:08.010 --> 00:20:11.345
you have is mu equals
0, sigma equals 1,
00:20:11.345 --> 00:20:13.011
if you want to write
it in this formula.
00:20:19.460 --> 00:20:25.940
What you're going to have
is 2 times B_t of dB_t
00:20:25.940 --> 00:20:27.490
plus 1 over 2 times 2dt.
00:20:30.322 --> 00:20:31.266
If you write it down.
00:20:34.570 --> 00:20:36.815
You can either use
these parameters
00:20:36.815 --> 00:20:41.200
and just plug in each of
them to figure it out.
00:20:41.200 --> 00:20:43.130
Or a different way
to do it is really
00:20:43.130 --> 00:20:45.340
write down, remember the proof.
00:20:45.340 --> 00:20:48.490
This is partial f
over partial t dt
00:20:48.490 --> 00:20:58.350
plus partial f over partial x
dx plus 1/2-- remember this one.
00:20:58.350 --> 00:21:00.045
And x is dB_t here.
00:21:04.190 --> 00:21:09.100
That one is 0, that one
was 2x, so 2B_t dB_t.
00:21:09.100 --> 00:21:11.872
Use it one more
time, so you get dt.
00:21:20.600 --> 00:21:21.280
Make sense?
00:21:24.160 --> 00:21:26.855
Let's do a few more examples.
00:22:03.150 --> 00:22:06.876
And you want to compute
d of f at t comma B of t.
00:22:11.280 --> 00:22:13.810
Let's do it this time.
00:22:13.810 --> 00:22:19.440
Again, partial f over
partial t dt plus partial f
00:22:19.440 --> 00:22:23.500
over partial x dB_t.
00:22:23.500 --> 00:22:24.820
That's the first-order terms.
00:22:24.820 --> 00:22:28.740
The second-order term
is 1/2 partial square f
00:22:28.740 --> 00:22:35.728
over partial x square of dB_t
square, which is equal to dt.
00:22:43.140 --> 00:22:43.640
Let's do it.
00:22:43.640 --> 00:22:48.720
Partial f over partial
t, you get mu times f.
00:22:48.720 --> 00:22:51.016
This one is just
equal to mu times f.
00:22:53.720 --> 00:22:55.220
Maybe I'm going too quick.
00:22:55.220 --> 00:23:02.510
Mu times e to the
mu t plus dx, dt.
00:23:02.510 --> 00:23:05.593
Partial f over partial
x is sigma times e
00:23:05.593 --> 00:23:12.120
to the mu t plus
dx, and then dB_t
00:23:12.120 --> 00:23:15.190
plus-- if you take
the second derivative,
00:23:15.190 --> 00:23:17.545
you do that again,
what you get is
00:23:17.545 --> 00:23:25.872
1/2, and then sigma square
times e to the mu t plus dx, dt.
00:23:25.872 --> 00:23:26.372
Yes?
00:23:26.372 --> 00:23:28.012
AUDIENCE: In the original
equation that you just wrote,
00:23:28.012 --> 00:23:29.816
isn't it 1/2 times
sigma squared,
00:23:29.816 --> 00:23:31.784
and then the second derivative?
00:23:31.784 --> 00:23:33.854
Up there.
00:23:33.854 --> 00:23:34.520
PROFESSOR: Here?
00:23:34.520 --> 00:23:35.470
AUDIENCE: Yes.
00:23:35.470 --> 00:23:36.094
PROFESSOR: 1/2?
00:23:36.094 --> 00:23:37.940
AUDIENCE: Times sigma squared.
00:23:37.940 --> 00:23:40.510
PROFESSOR: Oh, sigma-- OK,
that's a good question.
00:23:40.510 --> 00:23:44.070
But that sigma is different.
00:23:44.070 --> 00:23:46.250
That's if you plug in X_t here.
00:23:46.250 --> 00:23:50.790
If you plug in X_t
where X_t is equal to mu
00:23:50.790 --> 00:23:57.170
prime dt plus sigma
prime d of B_t,
00:23:57.170 --> 00:23:59.500
then that sigma prime will
become a sigma prime square
00:23:59.500 --> 00:24:02.020
here.
00:24:02.020 --> 00:24:04.124
But here the function
is mu and sigma,
00:24:04.124 --> 00:24:05.540
so maybe it's not
a good notation.
00:24:05.540 --> 00:24:07.383
Let me use a and b here instead.
00:24:11.170 --> 00:24:13.890
The sigma here is
different from here.
00:24:13.890 --> 00:24:15.950
AUDIENCE: Yeah, that
makes a lot more sense.
00:24:15.950 --> 00:24:18.826
PROFESSOR: If you
replace a and b,
00:24:18.826 --> 00:24:23.310
but I already wrote down
all mu's and sigma's.
00:24:23.310 --> 00:24:25.495
That's a good point, actually.
00:24:25.495 --> 00:24:26.995
But that's when you
want to consider
00:24:26.995 --> 00:24:29.250
a general stochastic
process here
00:24:29.250 --> 00:24:31.276
other than Brownian motion.
00:24:31.276 --> 00:24:33.030
But here it's just
a Brownian motion,
00:24:33.030 --> 00:24:35.376
so it's the most simple form.
00:24:35.376 --> 00:24:36.375
And that's what you get.
00:24:39.300 --> 00:24:45.940
Mu plus 1/2 sigma square-- and
these are just all f itself.
00:24:45.940 --> 00:24:48.130
That's the good thing
about exponential.
00:24:48.130 --> 00:24:51.939
f times dt plus
sigma times d of B_t.
00:25:03.930 --> 00:25:04.570
Make sense?
00:25:14.180 --> 00:25:17.896
And there's a reason I
was covering this example.
00:25:17.896 --> 00:25:22.140
It's because-- let's come
back to this question.
00:25:22.140 --> 00:25:35.110
You want to model stock price
using Brownian motion, Brownian
00:25:35.110 --> 00:25:40.760
process, S of t.
00:25:40.760 --> 00:25:43.040
But you don't want S_t
to be a Brownian motion.
00:25:43.040 --> 00:25:46.760
What you want is a
percentile difference
00:25:46.760 --> 00:25:51.400
to be a Brownian motion, so you
want this percentile difference
00:25:51.400 --> 00:25:58.505
to behave like a Brownian
motion with some variance.
00:26:03.350 --> 00:26:11.500
The question was, is S_t equal
to e to the sigma times B of t
00:26:11.500 --> 00:26:12.745
in this case?
00:26:12.745 --> 00:26:16.260
And I already told you last
time that no, it's not true.
00:26:16.260 --> 00:26:18.360
We can now see
why it's not true.
00:26:18.360 --> 00:26:20.880
Take this function, S_t
equals e to the sigma
00:26:20.880 --> 00:26:24.990
B_t, that's exactly where
mu is equal to 0 here.
00:26:24.990 --> 00:26:30.590
What we got here was d of S_t,
in this case, is equal to-- mu
00:26:30.590 --> 00:26:36.430
is 0, so we get 1/2 of sigma
square times dt plus sigma
00:26:36.430 --> 00:26:37.274
times d of B_t.
00:26:40.180 --> 00:26:44.670
We originally were
targeting sigma times dB_t,
00:26:44.670 --> 00:26:47.890
but we got this
additional term which we
00:26:47.890 --> 00:26:51.112
didn't want in the first hand.
00:26:51.112 --> 00:26:52.570
In other words, we
have this drift.
00:27:01.455 --> 00:27:03.080
I wasn't really clear
in the beginning,
00:27:03.080 --> 00:27:05.850
but our goal was to
model stock price
00:27:05.850 --> 00:27:10.920
where the expected
value is 0 at all times.
00:27:10.920 --> 00:27:12.970
Our guess was to take
e to the sigma B_t,
00:27:12.970 --> 00:27:14.890
but it turns out
that in this case
00:27:14.890 --> 00:27:16.870
we have a drift,
if you just take
00:27:16.870 --> 00:27:19.570
natural e to the sigma of B_t.
00:27:19.570 --> 00:27:21.230
To remove that drift,
what you can do
00:27:21.230 --> 00:27:23.350
is subtract that term somehow.
00:27:23.350 --> 00:27:26.570
If you can get rid of that
term then you can see,
00:27:26.570 --> 00:27:30.120
if you add this mu to be
minus 1 over 2 sigma square,
00:27:30.120 --> 00:27:31.340
you can remove that term.
00:27:34.460 --> 00:27:35.650
That's why it doesn't work.
00:27:35.650 --> 00:27:47.570
So instead use S of t equals
e to the minus 1 over 2 sigma
00:27:47.570 --> 00:27:52.532
square t plus sigma of B_t.
00:27:58.330 --> 00:28:02.850
That's the geometric Brownian
motion without drift.
00:28:02.850 --> 00:28:05.820
And the reason it has no
drift is because of that.
00:28:05.820 --> 00:28:07.300
If you actually do
the computation,
00:28:07.300 --> 00:28:08.390
the dt term disappears.
00:28:28.911 --> 00:28:29.410
Question?
00:28:35.580 --> 00:28:39.000
So far we have been
discussing differentiation.
00:28:39.000 --> 00:28:40.706
Now let's talk
about integration.
00:28:40.706 --> 00:28:41.206
Yes?
00:28:41.206 --> 00:28:47.122
AUDIENCE: Could you we do get
this solution as [INAUDIBLE].
00:28:47.122 --> 00:28:50.573
Could you also
describe what it means?
00:28:50.573 --> 00:28:55.996
What does it mean,
this solution of B_t?
00:28:55.996 --> 00:28:58.461
Does that mean if we
have a sample path B_t,
00:28:58.461 --> 00:29:01.440
then we could get a
sample path for S_t?
00:29:01.440 --> 00:29:03.670
PROFESSOR: Oh, what
this means, yes.
00:29:03.670 --> 00:29:07.360
Whenever you have the B_t
value, just at each time
00:29:07.360 --> 00:29:10.050
take the exponential value.
00:29:10.050 --> 00:29:13.460
Because-- why we want to express
this in terms of a Brownian
00:29:13.460 --> 00:29:14.940
motion is, for
Brownian motion we
00:29:14.940 --> 00:29:17.030
have a pretty good
understanding.
00:29:17.030 --> 00:29:21.280
It's a really good process
you understand fairly well,
00:29:21.280 --> 00:29:25.200
and you have good control on it.
00:29:25.200 --> 00:29:28.840
But the problem is you want to
have a process whose percentile
00:29:28.840 --> 00:29:31.990
difference behaves
like a Brownian motion.
00:29:31.990 --> 00:29:34.340
And this gives you a
way of describing it
00:29:34.340 --> 00:29:37.290
in terms of Brownian motion, as
an exponential function of it.
00:29:43.390 --> 00:29:46.555
Does that answer your question?
00:29:46.555 --> 00:29:48.330
AUDIENCE: Right,
distribution means
00:29:48.330 --> 00:29:50.720
that if we have a
sample path B_t,
00:29:50.720 --> 00:29:54.544
that would be the corresponding
sample path for S of t?
00:29:54.544 --> 00:29:58.260
Is it a pointwise evaluation?
00:29:58.260 --> 00:30:00.270
PROFESSOR: That's a
good question, actually.
00:30:00.270 --> 00:30:02.950
Think of it as a
pointwise evaluation.
00:30:02.950 --> 00:30:07.400
That is not always
correct, but for most
00:30:07.400 --> 00:30:09.150
of the things that
we will cover,
00:30:09.150 --> 00:30:13.120
it's safe to think
about it that way.
00:30:13.120 --> 00:30:16.690
But if you think about it
path-wise all the time,
00:30:16.690 --> 00:30:18.680
eventually it fails.
00:30:18.680 --> 00:30:20.187
But that's a very
advanced topic.
00:30:32.130 --> 00:30:33.740
So what this question
is, basically
00:30:33.740 --> 00:30:37.350
B_t is a probability space,
it's a probability distribution
00:30:37.350 --> 00:30:39.390
over paths.
00:30:39.390 --> 00:30:43.060
For this equation, if you just
look at it, it looks right,
00:30:43.060 --> 00:30:44.650
but it doesn't
really make sense,
00:30:44.650 --> 00:30:46.964
because B_t-- if it's a
probability distribution, what
00:30:46.964 --> 00:30:47.630
is e to the B_t?
00:30:50.450 --> 00:30:52.530
Basically, what
it's saying is B_t
00:30:52.530 --> 00:30:55.210
is a probability
distribution over paths.
00:30:55.210 --> 00:30:58.700
If you take omega according
to-- a path according
00:30:58.700 --> 00:31:02.890
to the Brownian motion sample
probability distribution,
00:31:02.890 --> 00:31:08.230
and for this path it's well
defined, this function.
00:31:08.230 --> 00:31:13.700
So the probability density
function of this path
00:31:13.700 --> 00:31:16.910
is equal to the probability
density function of e
00:31:16.910 --> 00:31:19.435
to the whatever that is
in this distribution.
00:31:24.410 --> 00:31:26.300
Maybe it confused you more.
00:31:26.300 --> 00:31:30.009
Just consider this as some path,
some well-defined function,
00:31:30.009 --> 00:31:31.550
and you have a
well-defined function.
00:31:39.490 --> 00:31:40.853
Integral definition.
00:31:44.000 --> 00:31:46.510
I will first give you a
very, very stupid definition
00:31:46.510 --> 00:31:49.340
of integration.
00:31:49.340 --> 00:32:00.293
We say that we define
F as the integration...
00:32:12.270 --> 00:32:25.574
if d of F is equal
to f dB_t plus-- we
00:32:25.574 --> 00:32:27.365
define it as an inverse
of differentiation.
00:32:30.200 --> 00:32:34.860
Because differentiation
is now well-defined--
00:32:34.860 --> 00:32:39.690
we just defined integration
as the inverse of it,
00:32:39.690 --> 00:32:42.170
just as in classical calculus.
00:32:46.030 --> 00:32:47.810
So far, it doesn't
have that good meaning,
00:32:47.810 --> 00:32:51.160
other than being
an inverse of it,
00:32:51.160 --> 00:32:52.710
but at least it's well-defined.
00:32:52.710 --> 00:32:54.780
The question is, does it exist?
00:32:54.780 --> 00:32:57.445
Given f and g, does it exist,
does integration always exist,
00:32:57.445 --> 00:32:58.060
and so on.
00:32:58.060 --> 00:32:59.690
There's lots of
questions to ask,
00:32:59.690 --> 00:33:02.570
but at least this
is some definition.
00:33:02.570 --> 00:33:11.760
And the natural question is,
does there exist a Riemannian
00:33:11.760 --> 00:33:12.740
sum type description?
00:33:25.430 --> 00:33:28.580
That means-- if you remember
how we defined integral
00:33:28.580 --> 00:33:47.740
in calculus, you have a
function f, integration
00:33:47.740 --> 00:33:58.570
of f from a to b according to
the Riemannian sum description
00:33:58.570 --> 00:34:04.660
was, you just chop the
interval into very fine pieces,
00:34:04.660 --> 00:34:07.730
a_0, a_1, a_2, a_3,
dot, dot, dot--
00:34:07.730 --> 00:34:16.750
and then sum the area of these
boxes, and take the limit.
00:34:16.750 --> 00:34:20.570
And this is the limit
of Riemannian sums.
00:34:26.420 --> 00:34:33.920
Slightly more, if you want, it's
the limit as n goes to infinity
00:34:33.920 --> 00:34:40.790
of the function 1 over n times
the sum of i equal zero to t--
00:34:40.790 --> 00:34:48.587
I'll just call it 0 to b-- f of
t*b over n minus f of t minus 1
00:34:48.587 --> 00:34:51.461
over n.
00:34:51.461 --> 00:34:52.734
Does this ring a bell?
00:35:03.787 --> 00:35:04.287
Question?
00:35:04.287 --> 00:35:05.162
AUDIENCE: [INAUDIBLE]
00:35:10.780 --> 00:35:13.832
PROFESSOR: No, you're right.
00:35:13.832 --> 00:35:17.115
Good point, no we don't.
00:35:17.115 --> 00:35:17.615
Thanks.
00:35:22.570 --> 00:35:26.010
Does integral
defined in this way
00:35:26.010 --> 00:35:31.390
have this Riemannian sum type
description, is the question.
00:35:31.390 --> 00:35:33.110
So keep that in mind.
00:35:33.110 --> 00:35:36.660
I will come back to
this point later.
00:35:36.660 --> 00:35:41.839
In fact, it turns out to
be a very deep question
00:35:41.839 --> 00:35:43.630
and very important
question, this question,
00:35:43.630 --> 00:35:48.850
because if you remember
like I hope you remember,
00:35:48.850 --> 00:35:51.860
in the Riemannian sum, it
didn't matter which point you
00:35:51.860 --> 00:35:54.610
took in this interval.
00:35:54.610 --> 00:35:56.910
That was the whole point.
00:35:56.910 --> 00:35:58.780
You have the function.
00:35:58.780 --> 00:36:02.430
In the interval a_i to
a_(i+1), you take any point
00:36:02.430 --> 00:36:07.610
in the middle and make a
rectangle according to that
00:36:07.610 --> 00:36:08.642
point.
00:36:08.642 --> 00:36:10.350
And then, no matter
which point you take,
00:36:10.350 --> 00:36:12.740
when you go to the limit,
you had exactly the same sum
00:36:12.740 --> 00:36:14.190
all the time.
00:36:14.190 --> 00:36:16.560
That's how you define the limit.
00:36:16.560 --> 00:36:20.960
But what's really
interesting here
00:36:20.960 --> 00:36:24.090
is that it's no longer true.
00:36:24.090 --> 00:36:26.000
If you take the left
point all the time,
00:36:26.000 --> 00:36:28.110
and you take the right
point all the time,
00:36:28.110 --> 00:36:30.570
the two limits are different.
00:36:30.570 --> 00:36:33.020
And again, that's due to
the quadratic variation,
00:36:33.020 --> 00:36:38.980
because that much of variance
can accumulate over time.
00:36:38.980 --> 00:36:42.560
That's the reason we didn't
start with Riemannian sum type
00:36:42.560 --> 00:36:44.450
definition of integral.
00:36:44.450 --> 00:36:47.490
But I'll just make one remark.
00:36:47.490 --> 00:37:01.220
Ito integral is the
limit of Riemannian sums
00:37:01.220 --> 00:37:08.240
when always take the leftmost
point of each interval.
00:37:16.700 --> 00:37:20.670
So you chop down this curve
at-- the time interval
00:37:20.670 --> 00:37:23.106
into pieces, and
for each rectangle,
00:37:23.106 --> 00:37:25.230
pick the leftmost point,
and use it as a rectangle.
00:37:30.722 --> 00:37:31.680
And you take the limit.
00:37:31.680 --> 00:37:33.570
That will be your
Ito integral defined.
00:37:33.570 --> 00:37:37.070
It will be exactly equal to this
thing, the inverse of our Ito
00:37:37.070 --> 00:37:38.754
differentiation.
00:37:38.754 --> 00:37:40.170
I won't be able
to go into detail.
00:37:40.170 --> 00:37:44.080
What's more
interesting is instead,
00:37:44.080 --> 00:37:47.170
what happens if you take the
rightmost point all the time,
00:37:47.170 --> 00:37:51.680
you get an equivalent
theory of calculus.
00:37:51.680 --> 00:37:53.380
It's just like Ito's calculus.
00:37:53.380 --> 00:37:57.160
It looks really, really similar
and it's coherent itself,
00:37:57.160 --> 00:37:59.170
so there is no
logical flaw in it.
00:37:59.170 --> 00:38:01.260
It all makes sense,
but the only difference
00:38:01.260 --> 00:38:03.984
is instead of a plus in
the second-order term,
00:38:03.984 --> 00:38:04.650
you get minuses.
00:38:07.500 --> 00:38:09.940
Let me just make this
remark, because it's just
00:38:09.940 --> 00:38:15.350
a theoretical part, this thing,
but I think it's really cool.
00:38:15.350 --> 00:38:22.150
Remark-- there's this
and equivalent version.
00:38:22.150 --> 00:38:24.430
Maybe equivalent is
not the right word,
00:38:24.430 --> 00:38:26.820
but a very similar
version of Ito
00:38:26.820 --> 00:38:33.000
calculus such that
basically, what
00:38:33.000 --> 00:38:38.510
it says is d B_t square
is equal to minus dt.
00:38:38.510 --> 00:38:40.320
Then that changed
a lot of things.
00:38:40.320 --> 00:38:44.510
But this part, it's
not that important.
00:38:44.510 --> 00:38:48.740
Just cool stuff.
00:38:48.740 --> 00:38:53.410
Let's think about this a
little bit more, this fact.
00:38:53.410 --> 00:38:55.970
Taking the leftmost
point all the time
00:38:55.970 --> 00:38:59.820
means if you want to make
a decision for your time
00:38:59.820 --> 00:39:05.630
interval-- so at time t of
i and time t of i plus 1,
00:39:05.630 --> 00:39:08.760
let's say it's the stock price.
00:39:08.760 --> 00:39:14.590
You want to say that you had
so many stocks in this time
00:39:14.590 --> 00:39:16.370
interval.
00:39:16.370 --> 00:39:20.400
Let's say you had so many
stocks in this time interval
00:39:20.400 --> 00:39:23.430
according to the values
between this and this.
00:39:23.430 --> 00:39:26.660
In real world, your only
choice you have is you
00:39:26.660 --> 00:39:30.700
have to make the
decision at time t of i.
00:39:30.700 --> 00:39:33.580
Your choice cannot depend
on the future time.
00:39:33.580 --> 00:39:36.650
You can't suddenly say, OK,
in this interval the stock
00:39:36.650 --> 00:39:38.420
price increased a
lot, so I'll assume
00:39:38.420 --> 00:39:42.930
that I had a lot of
stocks in this interval.
00:39:42.930 --> 00:39:46.190
In this interval, I knew
it was going to drop,
00:39:46.190 --> 00:39:50.060
so I'll just take the
rightmost interval.
00:39:50.060 --> 00:39:52.410
I'll assume that I only
had this many stock.
00:39:52.410 --> 00:39:53.460
You can't do that.
00:39:53.460 --> 00:39:56.850
Your decision has to be
based on the leftmost point,
00:39:56.850 --> 00:39:58.630
because the time.
00:39:58.630 --> 00:40:00.880
You can't see the future.
00:40:00.880 --> 00:40:04.690
And the reason Ito's calculus
works well in our setting is
00:40:04.690 --> 00:40:09.380
because of this fact, because it
has inside it the fact that you
00:40:09.380 --> 00:40:10.820
cannot see the future.
00:40:10.820 --> 00:40:16.310
Every decision is made
based on the leftmost time.
00:40:16.310 --> 00:40:18.845
If you want to make a decision
for your time interval,
00:40:18.845 --> 00:40:21.700
you have to do it
in the beginning.
00:40:21.700 --> 00:40:27.240
That intuition is hidden
inside of the theory,
00:40:27.240 --> 00:40:29.390
and that's why it works so well.
00:40:29.390 --> 00:40:33.450
Let me reiterate this
part a little bit more.
00:40:33.450 --> 00:40:36.500
It's the definition
of these things
00:40:36.500 --> 00:40:39.540
where you're only
allowed to-- at time t,
00:40:39.540 --> 00:40:42.230
you're only allowed to use
the information up to time t.
00:40:54.390 --> 00:41:26.010
Definition: delta t is an
adapted process-- sorry,
00:41:26.010 --> 00:41:29.730
adapted to another
stochastic process X_t--
00:41:29.730 --> 00:41:38.920
if for all values
of time variables
00:41:38.920 --> 00:41:48.110
delta t depends only
on X_0 up to X_t.
00:41:50.930 --> 00:41:53.360
There's a lot of vague
statements inside here,
00:41:53.360 --> 00:41:55.120
but what I'm trying
to say is just
00:41:55.120 --> 00:41:59.947
assume X is the Brownian
motion underlying stock price.
00:41:59.947 --> 00:42:00.905
Your stock is changing.
00:42:04.500 --> 00:42:06.305
You want to come
up with a strategy,
00:42:06.305 --> 00:42:08.270
and you want to say
that mathematically
00:42:08.270 --> 00:42:11.280
this strategy makes sense.
00:42:11.280 --> 00:42:13.220
And what it's saying
is if your strategy
00:42:13.220 --> 00:42:17.050
makes your decision
at time t is only
00:42:17.050 --> 00:42:19.840
based on the past values
of your stock price,
00:42:19.840 --> 00:42:23.750
then that's an adapted process.
00:42:23.750 --> 00:42:26.590
This defines the processes
that are reasonable,
00:42:26.590 --> 00:42:28.540
that cannot see future.
00:42:28.540 --> 00:42:31.030
And these are all--
in terms of strategy,
00:42:31.030 --> 00:42:34.400
if delta_t is a
portfolio strategy,
00:42:34.400 --> 00:42:37.262
these are the only meaningful
strategies that you can use.
00:42:40.240 --> 00:42:42.740
And because of what I said
before, because we're always
00:42:42.740 --> 00:42:45.580
taking the leftmost
point, adaptive
00:42:45.580 --> 00:42:50.440
processes just also fit very
well with Ito's calculus.
00:42:50.440 --> 00:42:53.050
They'll come into
play altogether.
00:42:55.670 --> 00:42:56.655
Just a few examples.
00:43:10.445 --> 00:43:13.740
First, a very stupid example.
00:43:13.740 --> 00:43:15.290
X_t is adapted to X_t.
00:43:20.170 --> 00:43:23.230
Of course, because
at time, X_t really
00:43:23.230 --> 00:43:26.710
depends on only
X_t, nothing else.
00:43:26.710 --> 00:43:36.060
Two, X_(t+1) is
not adapted to X_t.
00:43:36.060 --> 00:43:37.580
This is maybe a
little bit vague,
00:43:37.580 --> 00:43:41.412
so we'll call it
Y_t equals X_(t+1).
00:43:44.090 --> 00:43:49.240
Y_t is the value at t
plus 1, and it's not based
00:43:49.240 --> 00:43:50.850
on the values up to time t.
00:43:50.850 --> 00:43:52.600
Just a very artificial example.
00:43:56.405 --> 00:44:03.178
Another example, delta
t equals minimum...
00:44:06.136 --> 00:44:07.122
is adapted.
00:44:21.419 --> 00:44:23.180
And I'll let you think about it.
00:44:23.180 --> 00:44:24.850
The fourth is quite interesting.
00:44:24.850 --> 00:44:27.540
Suppose T is fixed,
some large integer,
00:44:27.540 --> 00:44:30.140
or some large real number.
00:44:30.140 --> 00:44:44.170
Then you let delta t to be the
maximum where X of s, where...
00:44:50.600 --> 00:44:51.370
It's not adapted.
00:44:58.790 --> 00:44:59.455
What is this?
00:44:59.455 --> 00:45:02.020
This means at time T,
I'm going to take at it
00:45:02.020 --> 00:45:08.850
this value, the
maximum of all value
00:45:08.850 --> 00:45:11.340
inside this part, the future.
00:45:11.340 --> 00:45:13.469
This refers to the future.
00:45:13.469 --> 00:45:14.635
It's not an adapted process.
00:45:21.637 --> 00:45:22.220
Any questions?
00:45:25.290 --> 00:45:28.340
Now we're ready to talk
about the properties
00:45:28.340 --> 00:45:31.190
of Ito's integral.
00:45:31.190 --> 00:45:34.340
Let's quickly
review what we have.
00:45:34.340 --> 00:45:38.380
First, I defined Ito's lemma--
that means differentiation
00:45:38.380 --> 00:45:41.250
in Ito calculus.
00:45:41.250 --> 00:45:45.080
Then I defined integration using
differentiation-- integration
00:45:45.080 --> 00:45:48.020
was an inverse operation
of the differentiation.
00:45:48.020 --> 00:45:50.500
But this integration also had
an alternative description
00:45:50.500 --> 00:45:53.260
in terms of
Riemannian sums, where
00:45:53.260 --> 00:45:58.650
you're taking just the
leftmost point as the reference
00:45:58.650 --> 00:46:01.700
point for each interval.
00:46:01.700 --> 00:46:04.370
And then, as you
see, this naturally
00:46:04.370 --> 00:46:08.090
had this concept of
using the leftmost point.
00:46:08.090 --> 00:46:12.180
And to abstract
that concept, we've
00:46:12.180 --> 00:46:15.660
come up with this adapted
process, very natural process,
00:46:15.660 --> 00:46:17.710
which is like the
real-life procedures,
00:46:17.710 --> 00:46:20.900
real-life strategies
we can think of.
00:46:20.900 --> 00:46:22.700
Now let's see what
happens when you
00:46:22.700 --> 00:46:25.442
take the integral of
adapted processes.
00:46:25.442 --> 00:46:27.870
Ito integral has
really cool properties.
00:46:59.540 --> 00:47:03.000
The first thing is about
normal distribution.
00:47:03.000 --> 00:47:08.840
B_t has normal
distribution of 0 up to t.
00:47:08.840 --> 00:47:11.170
So your Brownian
motion at time t
00:47:11.170 --> 00:47:13.780
has normal
distribution with 0, t.
00:47:13.780 --> 00:47:17.090
That means if your stochastic
process is some constant times
00:47:17.090 --> 00:47:23.540
B of t, of course, then
you have 0 and c square t.
00:47:23.540 --> 00:47:26.780
It's still a normal variable.
00:47:26.780 --> 00:47:28.770
That means if you
integrate, that's
00:47:28.770 --> 00:47:31.130
the integration of some sigma.
00:47:39.878 --> 00:47:42.058
That's the integration
of sigma of dB_t.
00:47:46.680 --> 00:47:49.280
If sigma is a fixed
constant, when
00:47:49.280 --> 00:47:52.830
you take the Ito
integral of sigma times
00:47:52.830 --> 00:47:55.200
dB_t, this constant,
at each time
00:47:55.200 --> 00:47:58.210
you get a normal distribution.
00:47:58.210 --> 00:48:00.550
And this is like saying the
sum of normal distribution
00:48:00.550 --> 00:48:02.330
is also normal distribution.
00:48:02.330 --> 00:48:04.090
It has this hidden
fact, because integral
00:48:04.090 --> 00:48:06.980
is like sum in the limit.
00:48:06.980 --> 00:48:10.456
And this can be generalized.
00:48:10.456 --> 00:48:18.560
If delta t is a process
depending only on the time
00:48:18.560 --> 00:48:27.660
variable-- so it does not depend
on the Brownian motion-- then
00:48:27.660 --> 00:48:35.810
the process X of t equals the
integration of delta t dB_t
00:48:35.810 --> 00:48:50.420
has normal distribution at
all time, just like this.
00:48:50.420 --> 00:48:52.580
We don't know the
exact variance yet;
00:48:52.580 --> 00:48:55.280
the variance will
depend on the sigmas.
00:48:55.280 --> 00:48:57.334
But still, it's like a
sum of normal variables,
00:48:57.334 --> 00:48:58.750
so we'll have
normal distribution.
00:49:03.490 --> 00:49:05.360
In fact, it just gets
better and better.
00:49:10.140 --> 00:49:14.950
The second fact is
called Ito isometry.
00:49:14.950 --> 00:49:15.930
That was cool.
00:49:15.930 --> 00:49:17.136
Can we compute the variance?
00:49:29.611 --> 00:49:30.110
Yes?
00:49:30.110 --> 00:49:31.466
AUDIENCE: Can you
put that board up?
00:49:31.466 --> 00:49:32.132
PROFESSOR: Sure.
00:49:34.630 --> 00:49:35.970
AUDIENCE: Does it go up?
00:49:35.970 --> 00:49:37.900
PROFESSOR: This
one doesn't go up.
00:49:37.900 --> 00:49:40.070
That's bad.
00:49:40.070 --> 00:49:41.240
I wish it did go up.
00:49:49.020 --> 00:49:52.060
This has a name
called Ito isometry.
00:49:56.740 --> 00:49:58.890
Can be used to
compute the variance.
00:49:58.890 --> 00:50:01.640
B_t is a Brownian
motion, delta t
00:50:01.640 --> 00:50:03.480
is adapted to a Brownian motion.
00:50:09.050 --> 00:50:17.610
Then the expectation
of your Ito integral--
00:50:17.610 --> 00:50:21.600
that's the Ito integral
of your adapted process.
00:50:21.600 --> 00:50:25.440
That's the variance-- we
take the square of it--
00:50:25.440 --> 00:50:29.847
is equal to something cool.
00:50:36.180 --> 00:50:38.456
The square just comes in.
00:50:38.456 --> 00:50:39.500
Quite nice, isn't it?
00:50:44.520 --> 00:50:48.400
I won't prove it, but
let me tell you why.
00:50:48.400 --> 00:50:50.300
We already saw this
phenomenon before.
00:50:50.300 --> 00:50:51.960
This is basically
quadratic variation.
00:50:58.000 --> 00:50:59.560
And the proof also uses it.
00:50:59.560 --> 00:51:03.160
If you take delta s
equals to 1-- sorry,
00:51:03.160 --> 00:51:09.680
I was using Korean-- 1 at all
time, then what we have is
00:51:09.680 --> 00:51:13.490
here you get a
Brownian motion, B_t.
00:51:13.490 --> 00:51:19.530
So on the left you get like
expectation of B_t square,
00:51:19.530 --> 00:51:21.525
and on the right,
what you get is t.
00:51:24.440 --> 00:51:27.445
Because when delta
s is equal to 1
00:51:27.445 --> 00:51:30.260
at all time, when you have
to get from 0 to t you get t,
00:51:30.260 --> 00:51:32.730
and you have t on
the right hand side.
00:51:32.730 --> 00:51:35.180
That's what it's saying.
00:51:35.180 --> 00:51:37.455
And that was the content
of quadratic variation,
00:51:37.455 --> 00:51:38.840
if you remember.
00:51:38.840 --> 00:51:42.484
We're summing the squares--
maybe not exactly this,
00:51:42.484 --> 00:51:44.650
but you're summing the
squares over small intervals.
00:52:00.530 --> 00:52:02.510
So that's a really
good fact that you can
00:52:02.510 --> 00:52:05.900
use to compute the variance.
00:52:05.900 --> 00:52:08.220
You have an Ito integral,
you know the square,
00:52:08.220 --> 00:52:10.190
can be computed this simple way.
00:52:14.110 --> 00:52:15.090
That's really cool.
00:52:17.850 --> 00:52:19.080
And one more property.
00:52:19.080 --> 00:52:22.560
This one will be
really important.
00:52:22.560 --> 00:52:24.295
You'll see it a lot
in future lectures.
00:52:28.630 --> 00:52:31.190
It's that when is Ito
integral a martingale?
00:52:46.630 --> 00:52:48.030
What's a martingale?
00:52:48.030 --> 00:52:52.310
Martingale meant if you
have a stochastic process,
00:52:52.310 --> 00:53:01.080
at any time t, whatever happens
after that, the expected value
00:53:01.080 --> 00:53:03.890
at time t is equal to 0.
00:53:03.890 --> 00:53:07.630
It doesn't have any natural
tendency to go up or go down.
00:53:07.630 --> 00:53:10.357
No matter which point
you stop your process
00:53:10.357 --> 00:53:12.815
and you see your future, it
doesn't have a natural tendency
00:53:12.815 --> 00:53:15.470
to go up or go down.
00:53:15.470 --> 00:53:29.190
In formal language, it can
be defined as where F_t
00:53:29.190 --> 00:53:32.670
is the events X_0 up to X_t.
00:53:35.890 --> 00:53:39.610
So if you take the
conditional expectation
00:53:39.610 --> 00:53:42.300
based on whatever
happened up to time t,
00:53:42.300 --> 00:53:44.235
that expectation will
just be whatever value
00:53:44.235 --> 00:53:45.384
you have at that time.
00:53:48.524 --> 00:53:51.190
Intuitively, that just means you
don't have any natural tendency
00:53:51.190 --> 00:53:53.900
to go up or go down.
00:53:53.900 --> 00:53:59.470
Question is, when is an
Ito integral a martingale?
00:54:28.985 --> 00:54:35.710
Adapted to B of t, then...
00:54:45.344 --> 00:54:46.010
is a martingale.
00:54:51.030 --> 00:54:54.090
As long as g is not
some crazy function,
00:54:54.090 --> 00:55:05.392
as long as g is reasonable--
one way can be reasonable if its
00:55:05.392 --> 00:55:07.900
L^2-norm is bounded.
00:55:07.900 --> 00:55:11.540
If you don't know what it
means, you can safely ignore it.
00:55:19.030 --> 00:55:23.490
Basically, if g doesn't-- it's
not a crazy function if it
00:55:23.490 --> 00:55:27.800
doesn't grow too fast, then
in most cases this integral is
00:55:27.800 --> 00:55:28.960
always a martingale.
00:55:31.590 --> 00:55:34.800
If you flip it--
remember, integral
00:55:34.800 --> 00:55:38.880
was defined as the inverse
of differentiation.
00:55:38.880 --> 00:55:42.700
So if dX_t is equal to
some function mu, that
00:55:42.700 --> 00:55:48.631
depends on both t and
B_t, times dt, plus sigma
00:55:48.631 --> 00:55:56.940
of dB_t, what this means
is X_t is a martingale
00:55:56.940 --> 00:56:02.690
if that is 0 at
all time, always.
00:56:07.860 --> 00:56:09.410
And if it's not 0,
you have a drift,
00:56:09.410 --> 00:56:12.132
so it's not a martingale.
00:56:12.132 --> 00:56:13.590
That gives you some
classification.
00:56:13.590 --> 00:56:15.390
Now, if you look at a
differential equation
00:56:15.390 --> 00:56:17.310
of this stochastic--
this is called
00:56:17.310 --> 00:56:19.791
a stochastic differential
equation-- if you know
00:56:19.791 --> 00:56:22.290
stochastic process, if you look
at a stochastic differential
00:56:22.290 --> 00:56:26.440
equation, if it doesn't have a
drift term, it's a martingale.
00:56:26.440 --> 00:56:29.510
If it has a drift term,
it's not a martingale.
00:56:29.510 --> 00:56:32.310
That'll be really useful
later, so try to remember it.
00:56:32.310 --> 00:56:34.290
The whole point is
when you write down
00:56:34.290 --> 00:56:36.990
a stochastic process in
terms of something times dt,
00:56:36.990 --> 00:56:40.200
something times dB_t,
really this term
00:56:40.200 --> 00:56:45.640
contributes towards the
tendency, the slope of whatever
00:56:45.640 --> 00:56:47.320
is going to happen
in the future.
00:56:47.320 --> 00:56:50.595
And this is like
the variance term.
00:56:50.595 --> 00:56:54.430
It adds some variance to
your stochastic process.
00:56:54.430 --> 00:56:58.990
But still, it doesn't add
or subtract value over time,
00:56:58.990 --> 00:57:03.895
it fairly adds variation.
00:57:06.540 --> 00:57:07.150
Remember that.
00:57:07.150 --> 00:57:09.890
That's very important fact.
00:57:09.890 --> 00:57:11.900
You're going to use it a lot.
00:57:11.900 --> 00:57:14.430
For example, you're going to
use it for pricing theory.
00:57:14.430 --> 00:57:18.870
In pricing theory, you come up
with this stochastic process
00:57:18.870 --> 00:57:20.150
or some strategy.
00:57:20.150 --> 00:57:22.130
You look at its value.
00:57:22.130 --> 00:57:27.280
Let's say X_t is your value
of your portfolio over time.
00:57:27.280 --> 00:57:34.940
If that portfolio has-- then you
match it with your financial--
00:57:34.940 --> 00:57:36.830
let me go over it slowly again.
00:57:36.830 --> 00:57:44.780
First you have a financial
derivative, like option
00:57:44.780 --> 00:57:47.632
of a stock.
00:57:47.632 --> 00:57:49.215
Then you have your
portfolio strategy.
00:57:55.630 --> 00:57:57.720
Assume that you have
some strategy that,
00:57:57.720 --> 00:57:59.940
at the expiration
time, gives you
00:57:59.940 --> 00:58:01.310
the exact value of the option.
00:58:03.820 --> 00:58:05.820
Now you look at the
difference between these two
00:58:05.820 --> 00:58:06.695
stochastic processes.
00:58:10.940 --> 00:58:14.880
Basically what the thing is,
when your variance goes to 0,
00:58:14.880 --> 00:58:19.310
your drift also has to go to 0.
00:58:19.310 --> 00:58:20.920
So when you look
at the difference,
00:58:20.920 --> 00:58:24.010
if you can somehow get rid
of this variance term, that
00:58:24.010 --> 00:58:26.880
means no matter
what you do, that
00:58:26.880 --> 00:58:30.660
will govern the value
of your portfolio.
00:58:30.660 --> 00:58:34.084
If it's positive, that means
you can always make money,
00:58:34.084 --> 00:58:35.250
because there's no variance.
00:58:35.250 --> 00:58:37.280
Without variance,
you make money.
00:58:37.280 --> 00:58:41.770
That's called arbitrage,
and you cannot have that.
00:58:41.770 --> 00:58:43.980
But I won't go
into further detail
00:58:43.980 --> 00:58:46.870
because Vasily will
cover it next time.
00:58:46.870 --> 00:58:49.070
But just remember that flavor.
00:58:49.070 --> 00:58:51.820
So when you write something down
in a stochastic differential
00:58:51.820 --> 00:58:55.790
equation form, that
term is a drift term,
00:58:55.790 --> 00:58:57.272
that term is a variance term.
00:58:57.272 --> 00:58:59.230
And if you don't have
drift, it's a martingale.
00:59:01.876 --> 00:59:03.290
That is very important.
00:59:12.290 --> 00:59:12.950
Any questions?
00:59:12.950 --> 00:59:16.658
That's kind of the
basics of Ito calculus.
00:59:22.520 --> 00:59:26.260
I will give you some
exercises on it,
00:59:26.260 --> 00:59:29.520
mostly just basic computation
exercises, so that you'll
00:59:29.520 --> 00:59:31.120
get familiar with it.
00:59:31.120 --> 00:59:33.210
Try to practice it.
00:59:33.210 --> 00:59:38.320
And let me cover one more
thing called Girsanov theorem.
00:59:38.320 --> 00:59:40.900
It's related, but
these are really
00:59:40.900 --> 00:59:42.750
basics of the Ito
calculus, so if you
00:59:42.750 --> 00:59:46.300
have any questions on
this, please ask me
00:59:46.300 --> 00:59:48.724
right now before I move
on to the next topic.
00:59:56.842 --> 00:59:58.840
The last thing I want
to talk about today.
01:00:43.710 --> 01:00:47.007
Here is an underlying question.
01:00:47.007 --> 01:00:48.590
Suppose you have two
Brownian motions.
01:00:57.050 --> 01:00:58.240
This is without drift.
01:01:01.810 --> 01:01:06.905
And you have another B tilde,
Brownian motion with drift.
01:01:12.910 --> 01:01:15.530
These are two probability
distributions over paths.
01:01:18.290 --> 01:01:21.320
According to B_t, you're
more likely to have
01:01:21.320 --> 01:01:25.770
some Brownian motion
that has no drift.
01:01:25.770 --> 01:01:28.290
That's a sample path.
01:01:28.290 --> 01:01:31.090
According to B tilde,
you have some drift.
01:01:34.890 --> 01:01:37.533
Your Brownian motion will--
01:01:41.240 --> 01:01:46.190
A typical path will follow this
line and will follow that line.
01:01:46.190 --> 01:01:52.920
The question is
this-- can we switch
01:01:52.920 --> 01:01:55.490
from this distribution
to this distribution
01:01:55.490 --> 01:01:56.561
by a change of measure?
01:02:02.250 --> 01:02:12.730
Can we switch between
the two measures
01:02:12.730 --> 01:02:23.720
to probability distributions
by a change of measure?
01:02:30.990 --> 01:02:34.114
Let me go a little bit
more what it really means.
01:02:34.114 --> 01:02:36.280
Assume that you're just
looking at a Brownian motion
01:02:36.280 --> 01:02:41.760
from time 0 up to time t,
some fixed time interval.
01:02:41.760 --> 01:02:47.610
Then according to B_t, let's
say this is a sample path omega.
01:02:47.610 --> 01:02:54.060
You have some probability
of omega-- this is a p.d.f.
01:02:54.060 --> 01:03:01.830
given by this Brownian
motion B. And then you
01:03:01.830 --> 01:03:06.100
have another p.d.f., P tilde
of omega, which is a p.d.f.
01:03:06.100 --> 01:03:11.240
given by B of t.
01:03:11.240 --> 01:03:14.990
The question is,
does there exist a Z
01:03:14.990 --> 01:03:19.610
depending on omega
such that P of omega
01:03:19.610 --> 01:03:23.740
is equal to Z times P tilde?
01:03:39.298 --> 01:03:40.589
Do you understand the question?
01:03:46.220 --> 01:03:49.420
Clearly, if you just look at
it, they're quite different.
01:03:49.420 --> 01:03:52.220
The path that you get
according to the distributions
01:03:52.220 --> 01:03:55.230
are quite different.
01:03:55.230 --> 01:03:57.740
It's not clear why we
should expect it at all.
01:04:02.866 --> 01:04:03.990
You'll see the answer soon.
01:04:03.990 --> 01:04:07.260
But let me discuss all this
in a different context.
01:04:17.080 --> 01:04:19.880
Just forget about all the
Brownian motion and everything
01:04:19.880 --> 01:04:22.410
just for a moment.
01:04:22.410 --> 01:04:26.010
In this concept, changing from
one probability distribution
01:04:26.010 --> 01:04:29.270
to another distribution,
it's a very important concept
01:04:29.270 --> 01:04:32.930
in analysis and probability
just in general, theoretically.
01:04:32.930 --> 01:04:39.060
And there's a name for this
Z, for this changing measure.
01:04:39.060 --> 01:04:46.380
If Z exists, it's called the
Radon-Nikodym derivative.
01:04:50.900 --> 01:04:53.096
Before doing that, let me
talk a little bit more.
01:04:59.940 --> 01:05:03.910
Suppose P is a probability
distribution over omega.
01:05:09.204 --> 01:05:10.537
It's a probability distribution.
01:05:18.660 --> 01:05:21.560
So this is some set, and P
describes the probability
01:05:21.560 --> 01:05:25.660
that you have each
element in the set.
01:05:25.660 --> 01:05:27.970
And you have another probability
distribution, P tilde.
01:05:33.520 --> 01:05:45.760
We define P and P tilde to be
equivalent if the probability
01:05:45.760 --> 01:05:50.667
that A is greater than
zero if and only if...
01:05:53.800 --> 01:05:54.400
For all...
01:05:58.310 --> 01:06:01.150
These probability distributions
describe the probability
01:06:01.150 --> 01:06:03.380
of the subsets.
01:06:03.380 --> 01:06:05.210
Think about a very simple case.
01:06:05.210 --> 01:06:09.620
Sigma is equal to 1, 2, and 3.
01:06:09.620 --> 01:06:13.495
P gives 1/3 probability
to 1, 1/3 probability
01:06:13.495 --> 01:06:16.970
to 2, 1/3 probability to 3.
01:06:16.970 --> 01:06:22.660
P tilde gives 2/3 probability
to 3, 1 over 6 probability
01:06:22.660 --> 01:06:26.570
to 2, 1 over 6 probability to 3.
01:06:26.570 --> 01:06:29.290
We have two probability
distribution over some space.
01:06:32.020 --> 01:06:34.790
They are equivalent
if, whenever you
01:06:34.790 --> 01:06:39.210
take a subset of your
background set-- let's say 1, 2.
01:06:39.210 --> 01:06:41.560
When A is equal
to 1, 2, according
01:06:41.560 --> 01:06:44.810
to probability distribution
P, the probability
01:06:44.810 --> 01:06:48.125
you fall into this
set A is equal to 2/3.
01:06:50.770 --> 01:06:54.750
According to P
tilde, you have 5/6.
01:06:57.460 --> 01:06:58.590
They're not the same.
01:06:58.590 --> 01:07:00.460
The probability itself
is not the same,
01:07:00.460 --> 01:07:03.220
but this condition is
satisfied when it's 0.
01:07:03.220 --> 01:07:04.865
And when it's not 0, it's not 0.
01:07:04.865 --> 01:07:07.406
And you can just check that it's
always true, because they're
01:07:07.406 --> 01:07:09.360
all positive probabilities.
01:07:09.360 --> 01:07:14.270
On the other hand, if
you take instead, say,
01:07:14.270 --> 01:07:19.640
1/3 and 0, now you
take your A to be 3.
01:07:22.850 --> 01:07:28.610
Then you have 1/3 equal to 0.
01:07:28.610 --> 01:07:33.160
This means, according to
probability distribution P,
01:07:33.160 --> 01:07:37.320
there is some probability
that you'll get 3.
01:07:37.320 --> 01:07:39.990
But according to probability
distribution P tilde,
01:07:39.990 --> 01:07:43.970
you don't have any
probability of getting 3.
01:07:43.970 --> 01:07:47.620
So they're not
equivalent in this case.
01:07:47.620 --> 01:07:49.840
If you think about it,
then it's really clear.
01:07:49.840 --> 01:07:52.360
The theorem says-- this is
a very important theorem
01:07:52.360 --> 01:07:53.690
in analysis, actually.
01:07:57.875 --> 01:08:08.475
The theorem-- there exists a Z
such that P of omega is equal
01:08:08.475 --> 01:08:08.975
to...
01:08:12.380 --> 01:08:16.638
If and only if P and P
tilde are equivalent.
01:08:22.510 --> 01:08:24.750
You can change from
one probability measure
01:08:24.750 --> 01:08:27.029
to another probability
measure just
01:08:27.029 --> 01:08:32.231
in terms of multiplication, if
and only if they're equivalent.
01:08:32.231 --> 01:08:34.189
And you can see that it's
not the case for this
01:08:34.189 --> 01:08:35.355
when they're not equivalent.
01:08:35.355 --> 01:08:37.740
You can't make a zero
probability to 1/3 probability
01:08:37.740 --> 01:08:40.000
by multiplication.
01:08:40.000 --> 01:08:44.510
So in the finite world this is
very just intuitive theorem,
01:08:44.510 --> 01:08:48.490
but what this is saying is
it's true for all probability
01:08:48.490 --> 01:08:50.736
spaces.
01:08:50.736 --> 01:08:52.819
And these are called the
Radon-Nikodym derivative.
01:09:01.930 --> 01:09:06.990
Our question is, are these two
Brownian motions equivalent?
01:09:06.990 --> 01:09:09.915
The paths that this Brownian
motion without drift
01:09:09.915 --> 01:09:12.330
takes and the Brownian
motion with drift
01:09:12.330 --> 01:09:16.529
takes, are they kind of
the same but just skewed
01:09:16.529 --> 01:09:20.562
in distribution, or are they
really fundamentally different?
01:09:20.562 --> 01:09:21.538
That's the question.
01:09:28.859 --> 01:09:33.479
And what Girsanov's theorem says
is that they are equivalent.
01:09:33.479 --> 01:09:36.089
To me, it came as a
little bit non-intuitive.
01:09:36.089 --> 01:09:39.880
I would imagine that it's
not equivalent, these two.
01:09:39.880 --> 01:09:42.069
These paths have a
very natural tendency.
01:09:42.069 --> 01:09:44.870
As it goes to infinity,
these paths and these paths
01:09:44.870 --> 01:09:47.779
will really look
a lot different,
01:09:47.779 --> 01:09:51.000
because when you go
really, really far,
01:09:51.000 --> 01:09:53.939
the paths which have
drift will be just really
01:09:53.939 --> 01:09:57.870
close to your line mu of
t, while the paths which
01:09:57.870 --> 01:10:00.302
don't have drift will be
really close to the x-axis.
01:10:02.900 --> 01:10:06.070
But still, they are equivalent.
01:10:06.070 --> 01:10:09.590
You can change from
one to another.
01:10:09.590 --> 01:10:13.840
I'll just state that
theorem without proof.
01:10:13.840 --> 01:10:17.345
And this will also be
used in pricing theory.
01:10:20.930 --> 01:10:23.270
I'm not an expert
enough to tell why,
01:10:23.270 --> 01:10:25.140
but basically what
it's saying is,
01:10:25.140 --> 01:10:28.580
you switch some
stochastic process
01:10:28.580 --> 01:10:31.000
into a stochastic
process without drift,
01:10:31.000 --> 01:10:33.610
thus making it
into a martingale.
01:10:33.610 --> 01:10:36.180
And martingale has a lot of
meaning in pricing theory,
01:10:36.180 --> 01:10:38.310
as you'll see.
01:10:38.310 --> 01:10:39.920
This also has application.
01:10:39.920 --> 01:10:42.030
That's why I'm trying to
cover it, although it's
01:10:42.030 --> 01:10:44.300
quite a technical theorem.
01:10:44.300 --> 01:10:46.985
Try to remember at least
a statement and the spirit
01:10:46.985 --> 01:10:48.700
of what it means.
01:10:48.700 --> 01:10:50.690
It just means these
two are equivalent,
01:10:50.690 --> 01:10:52.370
you can change
from one to another
01:10:52.370 --> 01:10:53.830
by a multiplicative function.
01:11:08.267 --> 01:11:09.850
Let me just state
it in a simple form.
01:11:12.615 --> 01:11:14.740
GUEST SPEAKER: If I could
just interject a comment.
01:11:14.740 --> 01:11:15.406
PROFESSOR: Sure.
01:11:15.406 --> 01:11:19.830
GUEST SPEAKER: With
these changes of measure,
01:11:19.830 --> 01:11:24.620
it turns out that all of these
theories with continuous time
01:11:24.620 --> 01:11:27.530
processes should have an
interpretation if you've
01:11:27.530 --> 01:11:30.720
discretized time,
and should consider
01:11:30.720 --> 01:11:34.140
sort of a finer and finer
discretization of the process.
01:11:34.140 --> 01:11:40.680
And with this change of measure,
if you consider problems
01:11:40.680 --> 01:11:45.840
in discrete stochastic
processes like random walks,
01:11:45.840 --> 01:11:52.160
basically how-- say if you're
gambling against a casino
01:11:52.160 --> 01:11:54.510
or against another
player, and you
01:11:54.510 --> 01:11:58.200
look at how your winnings
evolve as a random walk,
01:11:58.200 --> 01:11:59.780
depending on your
odds, your odds
01:11:59.780 --> 01:12:03.340
could be that you
will tend to lose.
01:12:03.340 --> 01:12:06.740
So there's basically
a drift in your wealth
01:12:06.740 --> 01:12:08.940
as this random process evolves.
01:12:08.940 --> 01:12:15.820
You can transform that process,
basically by taking out
01:12:15.820 --> 01:12:19.560
your expected losses,
to a process which
01:12:19.560 --> 01:12:22.830
has zero change in expectation.
01:12:22.830 --> 01:12:26.840
And so you can convert
these gambling problems
01:12:26.840 --> 01:12:32.020
where there's drift to a version
where the process, essentially,
01:12:32.020 --> 01:12:34.460
has no drift and
is a martingale.
01:12:34.460 --> 01:12:37.240
And the martingale theory in
stochastic process courses
01:12:37.240 --> 01:12:38.560
is very, very powerful.
01:12:38.560 --> 01:12:41.090
There's martingale
convergence theorems.
01:12:41.090 --> 01:12:44.180
So you know that the
limit of the martingale
01:12:44.180 --> 01:12:48.750
is-- there's a convergence
of the process,
01:12:48.750 --> 01:12:50.530
and that applies here as well.
01:12:55.026 --> 01:12:57.234
PROFESSOR: You will see some
surprising applications.
01:12:57.234 --> 01:12:59.594
GUEST SPEAKER: Yeah.
01:12:59.594 --> 01:13:03.515
PROFESSOR: And try to at
least digest the statement.
01:13:08.540 --> 01:13:12.340
When the guest speaker comes
and says by Girsanov theorem,
01:13:12.340 --> 01:13:15.660
they actually know what it is.
01:13:15.660 --> 01:13:16.410
There's a spirit.
01:13:20.190 --> 01:13:21.707
This is a very simple version.
01:13:21.707 --> 01:13:23.290
There's a lot of
complicated versions,
01:13:23.290 --> 01:13:24.874
but let me just do it.
01:13:30.570 --> 01:13:40.025
So P is a probability
distribution over paths
01:13:40.025 --> 01:13:41.900
from [0, T] to the infinity.
01:13:41.900 --> 01:13:48.860
What this means is just paths
from that-- stochastic process
01:13:48.860 --> 01:13:53.110
defined from time
0 to time T. These
01:13:53.110 --> 01:14:10.790
are paths defined by a
Brownian motion with drift mu.
01:14:10.790 --> 01:14:14.080
And then P tilde is a
probability distribution
01:14:14.080 --> 01:14:16.821
defined by Brownian
motion without drift.
01:14:22.600 --> 01:14:27.445
Then P and P tilde
are equivalent.
01:14:27.445 --> 01:14:29.320
Not only are they
equivalent, we can actually
01:14:29.320 --> 01:14:31.530
compute their
Radon-Nikodym derivative.
01:14:36.184 --> 01:14:44.378
And the Radon-Nikodym
derivative Z
01:14:44.378 --> 01:14:50.100
which is defined as T of--
which we denote like this
01:14:50.100 --> 01:14:51.180
has this nice form.
01:15:05.920 --> 01:15:08.150
That's a nice closed form.
01:15:08.150 --> 01:15:13.120
Let me just tell you a
few implications of this.
01:15:31.490 --> 01:15:35.790
Now, assume you have
some, let's say, value
01:15:35.790 --> 01:15:37.430
of your portfolio over time.
01:15:37.430 --> 01:15:40.230
That's the stochastic process.
01:15:40.230 --> 01:15:44.090
And you measure it according to
this probability distribution.
01:15:44.090 --> 01:15:45.930
Let's say it depends
on some stock price
01:15:45.930 --> 01:15:47.970
as the stock price is
modeled using a Brownian
01:15:47.970 --> 01:15:51.140
motion with drift.
01:15:51.140 --> 01:15:53.500
What this is saying
is, now, instead
01:15:53.500 --> 01:15:57.920
of computing this expectation
in your probability space--
01:15:57.920 --> 01:16:03.140
so this is defined over
the probability space P,
01:16:03.140 --> 01:16:06.510
our sigma-- (omega, P)
defined by this probability
01:16:06.510 --> 01:16:07.610
distribution.
01:16:07.610 --> 01:16:25.730
You can instead
compute it in-- you
01:16:25.730 --> 01:16:28.720
can compute as expectation in
a different probability space.
01:16:35.080 --> 01:16:38.430
You transform the problems
about Brownian motion with drift
01:16:38.430 --> 01:16:41.420
into a problem about Brownian
motion without a drift.
01:16:41.420 --> 01:16:43.170
And the reason I have
Z tilde instead of Z
01:16:43.170 --> 01:16:45.230
here is because I flipped.
01:16:45.230 --> 01:16:54.480
What you really should have is Z
tilde here as expectation of Z.
01:16:54.480 --> 01:16:59.380
If you want to use this Z.
01:16:59.380 --> 01:17:03.350
I don't expect you to really
be able to do computations
01:17:03.350 --> 01:17:06.650
and do that just by looking
at this theorem once.
01:17:06.650 --> 01:17:09.384
Just really trying to
digest what it means
01:17:09.384 --> 01:17:13.050
and understand the flavor of
it, that you can transform
01:17:13.050 --> 01:17:14.650
problems in one
probability space
01:17:14.650 --> 01:17:16.990
to another probability space.
01:17:16.990 --> 01:17:19.460
And you can actually do that
when the two distributions are
01:17:19.460 --> 01:17:22.700
defined by Brownian motions
when one has drift and one
01:17:22.700 --> 01:17:25.000
doesn't have a drift.
01:17:25.000 --> 01:17:27.700
How we're going
to use it is we're
01:17:27.700 --> 01:17:29.802
going to transform a
non-martingale process
01:17:29.802 --> 01:17:30.885
into a martingale process.
01:17:35.314 --> 01:17:36.730
When you change
into martingale it
01:17:36.730 --> 01:17:39.662
has very good physical
meanings to it.
01:17:43.450 --> 01:17:44.680
That's it for today.
01:17:44.680 --> 01:17:48.030
And you only have one more
math lecture remaining
01:17:48.030 --> 01:17:51.580
and maybe one or two
homeworks but if you have two,
01:17:51.580 --> 01:17:54.950
the second one
won't be that long.
01:17:54.950 --> 01:17:57.340
And you'll have a lot of
guest lectures, exciting guest
01:17:57.340 --> 01:18:00.990
lectures, so try
not to miss them.