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PROFESSOR: And today
it's me, back again.
00:00:25.540 --> 00:00:28.600
And we'll study continuous
types of stochastic processes.
00:00:28.600 --> 00:00:31.420
So far we were discussing
discrete time processes.
00:00:39.270 --> 00:00:41.990
We studied the basics like
variance, expectation,
00:00:41.990 --> 00:00:45.270
all this stuff-- moments,
moment generating function,
00:00:45.270 --> 00:00:52.160
and some important concepts for
Markov chains, and martingales.
00:01:01.260 --> 00:01:04.180
So I'm sure a lot of
you would have forgot
00:01:04.180 --> 00:01:06.660
about what martingale
and Markov chains were,
00:01:06.660 --> 00:01:09.680
but try to review this
before the next few lectures.
00:01:09.680 --> 00:01:11.880
Because starting
next week when we
00:01:11.880 --> 00:01:15.970
start discussing continuous
types of stochastic processes--
00:01:15.970 --> 00:01:16.490
not from me.
00:01:16.490 --> 00:01:18.740
You're not going to hear
martingale from me that much.
00:01:18.740 --> 00:01:21.230
But from people-- say,
outside speakers--
00:01:21.230 --> 00:01:23.840
they're going to use
this martingale concept
00:01:23.840 --> 00:01:25.590
to do pricing.
00:01:25.590 --> 00:01:28.250
So I will give you
some easy exercises.
00:01:28.250 --> 00:01:30.180
You will have some
problems on martingales.
00:01:30.180 --> 00:01:34.610
Just refer back to the notes
that I had like a month ago,
00:01:34.610 --> 00:01:35.460
and just review.
00:01:35.460 --> 00:01:37.120
It won't be difficult
problems, but try
00:01:37.120 --> 00:01:40.901
to make the concept comfortable.
00:01:40.901 --> 00:01:41.400
OK.
00:01:41.400 --> 00:01:46.320
And then Peter taught
some time series analysis.
00:01:46.320 --> 00:01:49.625
Time series is just the same
as discrete time process.
00:01:53.640 --> 00:02:04.600
And regression analysis, this
was all done on discrete time.
00:02:04.600 --> 00:02:09.162
That means the underlying space
was x_1, x_2, x_3, dot dot
00:02:09.162 --> 00:02:10.726
dot, x_t.
00:02:14.369 --> 00:02:16.410
But now we're going to
talk about continuous time
00:02:16.410 --> 00:02:17.800
processes.
00:02:17.800 --> 00:02:18.512
What are they?
00:02:18.512 --> 00:02:20.720
They're just a collection
of random variables indexed
00:02:20.720 --> 00:02:22.610
by time.
00:02:22.610 --> 00:02:24.520
But now the time
is a real variable.
00:02:24.520 --> 00:02:27.430
Here, time was just
in integer values.
00:02:27.430 --> 00:02:29.570
Here, we have real variable.
00:02:29.570 --> 00:02:35.750
So a stochastic process
develops over time,
00:02:35.750 --> 00:02:39.606
and the time variable
is continuous now.
00:02:39.606 --> 00:02:43.060
It doesn't necessarily mean
that the process itself
00:02:43.060 --> 00:02:46.815
is continuous-- it may as
well look like these jumps.
00:02:46.815 --> 00:02:48.690
It may as well have a
lot of jumps like this.
00:02:51.280 --> 00:02:52.950
It just means that
the underlying time
00:02:52.950 --> 00:02:54.850
variable is continuous.
00:02:54.850 --> 00:02:57.070
Whereas when it
was discrete time,
00:02:57.070 --> 00:03:00.160
you were only looking
at specific observations
00:03:00.160 --> 00:03:02.780
at some times.
00:03:02.780 --> 00:03:05.500
I'll draw it here.
00:03:05.500 --> 00:03:12.290
Discrete time looks
more like that.
00:03:15.170 --> 00:03:16.482
OK.
00:03:16.482 --> 00:03:18.470
So the first
difficulty when you try
00:03:18.470 --> 00:03:20.830
to understand continuous time
stochastic processes when
00:03:20.830 --> 00:03:23.215
you look at it is, how do
you describe the probability
00:03:23.215 --> 00:03:25.635
distribution?
00:03:25.635 --> 00:03:35.690
How to describe the
probability distribution?
00:03:40.430 --> 00:03:43.935
So let's go back to
discrete time processes.
00:03:43.935 --> 00:03:46.833
So the universal example
was a simple random walk.
00:03:52.630 --> 00:03:56.470
And if you remember, how we
described it was x_t minus
00:03:56.470 --> 00:04:04.650
x_(t-1), was either 1 or minus
1, probability half each.
00:04:04.650 --> 00:04:07.450
This was how we described it.
00:04:07.450 --> 00:04:10.290
And if you think about it,
this is a slightly indirect way
00:04:10.290 --> 00:04:11.740
of describing the process.
00:04:11.740 --> 00:04:13.790
You're not describing
the probability
00:04:13.790 --> 00:04:18.850
of this process following
this path, it's like a path.
00:04:18.850 --> 00:04:20.390
Instead what you're
doing is, you're
00:04:20.390 --> 00:04:24.850
describing the probability
of this event happening.
00:04:24.850 --> 00:04:28.335
From time t to t plus 1,
what is the probability
00:04:28.335 --> 00:04:30.220
that it will go down?
00:04:30.220 --> 00:04:33.760
And at each step you describe
the probability altogether,
00:04:33.760 --> 00:04:36.280
when you combine them, you get
the probability distribution
00:04:36.280 --> 00:04:39.170
over the process.
00:04:39.170 --> 00:04:41.930
But you can't do it for
continuous time, right?
00:04:41.930 --> 00:04:44.280
The time variable is
continuous so you can't just
00:04:44.280 --> 00:04:48.160
take intervals t
and interval t prime
00:04:48.160 --> 00:04:49.954
and describe the difference.
00:04:49.954 --> 00:04:52.620
If you want to do that, you have
to do it infinitely many times.
00:04:52.620 --> 00:04:54.939
You have to do it for
all possible values.
00:04:54.939 --> 00:04:56.105
That's the first difficulty.
00:05:00.400 --> 00:05:02.955
Actually, that's
the main difficulty.
00:05:08.540 --> 00:05:12.780
And how can we handle this?
00:05:18.060 --> 00:05:20.430
It's not an easy question.
00:05:20.430 --> 00:05:22.460
And you'll see a very
indirect way to handle it.
00:05:22.460 --> 00:05:26.040
It's somewhat in the
spirit of this thing.
00:05:26.040 --> 00:05:29.910
But it's not like you draw some
path to describe a probability
00:05:29.910 --> 00:05:32.890
density of this path.
00:05:32.890 --> 00:05:35.150
That's the omega.
00:05:35.150 --> 00:05:37.365
What is the probability
density at omega?
00:05:40.390 --> 00:05:42.480
Of course, it's not
a discrete variable
00:05:42.480 --> 00:05:46.020
so you have a probability
density function, not
00:05:46.020 --> 00:05:47.249
a probability mass function.
00:05:52.628 --> 00:05:56.362
In fact, can we
even write it down?
00:05:56.362 --> 00:05:57.820
You'll later see
that we won't even
00:05:57.820 --> 00:06:01.150
be able to write this down.
00:06:01.150 --> 00:06:04.340
So just have this
in mind and you'll
00:06:04.340 --> 00:06:06.090
see what I was trying to say.
00:06:09.150 --> 00:06:14.000
So finally, I get to talk
about Brownian processes,
00:06:14.000 --> 00:06:14.890
Brownian motion.
00:06:20.830 --> 00:06:24.360
Some outside speakers already
started talking about it.
00:06:24.360 --> 00:06:26.480
I wish I already
was able to cover it
00:06:26.480 --> 00:06:29.790
before they talked about it, but
you'll see a lot more from now.
00:06:29.790 --> 00:06:31.370
And let's see what
it actually is.
00:06:38.840 --> 00:06:40.852
So it's described
as the following,
00:06:40.852 --> 00:06:42.310
it actually follows
from a theorem.
00:06:44.920 --> 00:06:55.000
There exists a
probability distribution
00:06:55.000 --> 00:07:13.240
over the set of continuous
functions from positive reals
00:07:13.240 --> 00:07:21.973
to the reals such that
first, B(0) is always 0.
00:07:21.973 --> 00:07:26.360
So probability of B(0)
is equal to 0 is 1.
00:07:26.360 --> 00:07:29.770
Number two-- we call
this stationary.
00:07:32.790 --> 00:07:43.540
For all s and t,
B(t) minus B(s) has
00:07:43.540 --> 00:07:50.850
normal distribution with mean
0 and variance t minus s.
00:07:50.850 --> 00:07:53.080
And the third--
independent increment.
00:08:04.700 --> 00:08:19.410
That means if intervals
[s i, t i] are not overlapping,
00:08:19.410 --> 00:08:25.845
then B(t_i) minus
B(s_i) are independent.
00:08:31.160 --> 00:08:33.710
So it's actually
a theorem saying
00:08:33.710 --> 00:08:35.960
that there is some
strange probability
00:08:35.960 --> 00:08:40.530
distribution over the
continuous functions
00:08:40.530 --> 00:08:42.490
from positive reals--
non-negative reals--
00:08:42.490 --> 00:08:44.551
to the reals.
00:08:44.551 --> 00:08:48.457
So if you look at some
continuous function,
00:08:48.457 --> 00:08:50.540
this theorem gives you a
probability distribution.
00:08:50.540 --> 00:08:55.150
It describes the probability
of this path happening.
00:08:55.150 --> 00:08:56.400
It doesn't really describe it.
00:08:56.400 --> 00:08:58.608
It just says that there
exists some distribution such
00:08:58.608 --> 00:09:04.790
that it always starts at
0 and it's continuous.
00:09:04.790 --> 00:09:12.250
Second, the distribution for all
fixed s and t, the distribution
00:09:12.250 --> 00:09:17.150
of this difference is
normally distributed
00:09:17.150 --> 00:09:20.440
with mean 0 and variance
t minus s, which
00:09:20.440 --> 00:09:24.140
scales according to the time.
00:09:24.140 --> 00:09:28.320
And then third,
independent increment means
00:09:28.320 --> 00:09:31.850
what happened between
this interval, [s1, t1],
00:09:31.850 --> 00:09:37.450
and [s2, t2], this
part and this part,
00:09:37.450 --> 00:09:39.850
is independent as long as
intervals do not overlap.
00:09:45.050 --> 00:09:47.780
It sounds very similar to
the simple random walk.
00:09:47.780 --> 00:09:50.800
But the reason we have to do
this very complicated process
00:09:50.800 --> 00:09:53.500
is because the
time is continuous.
00:09:53.500 --> 00:09:57.260
You can't really describe at
each time what's happening.
00:09:57.260 --> 00:10:01.470
Instead, what you're describing
is over all possible intervals
00:10:01.470 --> 00:10:03.270
what's happening.
00:10:03.270 --> 00:10:06.530
When you have a fixed interval,
it describes the probability
00:10:06.530 --> 00:10:07.352
distribution.
00:10:07.352 --> 00:10:09.060
And then when you have
several intervals,
00:10:09.060 --> 00:10:12.421
as long as they don't
overlap, they're independent.
00:10:12.421 --> 00:10:12.920
OK?
00:10:12.920 --> 00:10:15.780
And then by this theorem,
we call this probability
00:10:15.780 --> 00:10:18.980
distribution a Brownian motion.
00:10:18.980 --> 00:10:27.820
So probability distribution,
the definition, distribution
00:10:27.820 --> 00:10:36.290
given by this theorem is
called the Brownian motion.
00:10:44.731 --> 00:10:46.230
That's why I'm
saying it's indirect.
00:10:46.230 --> 00:10:48.500
I'm not saying Brownian
motion is this probability
00:10:48.500 --> 00:10:49.200
distribution.
00:10:49.200 --> 00:10:53.140
It satisfies these conditions,
but we are reversing it.
00:10:53.140 --> 00:10:55.060
Actually, we have these
properties in mind.
00:10:55.060 --> 00:10:57.309
We're not sure if such a
probability distribution even
00:10:57.309 --> 00:10:58.690
exists or not.
00:10:58.690 --> 00:11:01.080
And actually this theorem
is very, very difficult.
00:11:01.080 --> 00:11:03.580
I don't know how to
prove it right now.
00:11:03.580 --> 00:11:06.170
I have to go through a book.
00:11:06.170 --> 00:11:09.432
And even graduate
probability courses
00:11:09.432 --> 00:11:11.640
usually don't cover it
because it's really technical.
00:11:14.610 --> 00:11:18.690
That means this just shows
how continuous time stochastic
00:11:18.690 --> 00:11:24.170
processes can be so much more
complicated than discrete time.
00:11:24.170 --> 00:11:29.170
Then why are you-- why are
we studying continuous time
00:11:29.170 --> 00:11:31.380
processes when it's
so complicated?
00:11:31.380 --> 00:11:33.700
Well, you'll see in
the next few lectures.
00:11:37.487 --> 00:11:38.070
Any questions?
00:11:41.290 --> 00:11:42.289
OK.
00:11:42.289 --> 00:11:44.080
So let's go through
this a little bit more.
00:11:44.080 --> 00:11:44.990
AUDIENCE: Excuse me.
00:11:44.990 --> 00:11:45.746
PROFESSOR: Yes.
00:11:45.746 --> 00:11:50.030
AUDIENCE: So when you talk about
the probability distribution,
00:11:50.030 --> 00:11:51.458
what's the underlying space?
00:11:51.458 --> 00:11:52.890
Is it the space of--
00:11:52.890 --> 00:11:54.810
PROFESSOR: Yes, that's
a very good question.
00:11:54.810 --> 00:11:59.250
The space is the space
of all functions.
00:11:59.250 --> 00:12:01.770
That means it's a space
of all possible paths,
00:12:01.770 --> 00:12:03.630
if you want to think
about it this way.
00:12:03.630 --> 00:12:06.090
Just think about
all possible ways
00:12:06.090 --> 00:12:09.880
your variable can
evolve over time.
00:12:09.880 --> 00:12:12.900
And for some fixed
drawing for this path,
00:12:12.900 --> 00:12:17.050
there's some probability
that this path will happen.
00:12:17.050 --> 00:12:21.540
It's not the probability spaces
that you have been looking at.
00:12:21.540 --> 00:12:26.750
It's not one point-- well,
a point is now a path.
00:12:26.750 --> 00:12:28.240
And your probability
distribution
00:12:28.240 --> 00:12:33.310
is given over paths,
not for a fixed point.
00:12:33.310 --> 00:12:35.940
And that's also a reason why
it makes it so complicated.
00:12:40.494 --> 00:12:41.160
Other questions?
00:12:44.880 --> 00:12:47.735
So the main thing you have to
remember-- well, intuitively
00:12:47.735 --> 00:12:49.170
you will just know it.
00:12:49.170 --> 00:12:53.930
But one thing you want to try
to remember is this property.
00:12:53.930 --> 00:12:58.610
As your time scales, what
happens between that interval
00:12:58.610 --> 00:13:01.910
is it's like a normal variable.
00:13:01.910 --> 00:13:05.360
So this is a collection of
a bunch of normal variables.
00:13:05.360 --> 00:13:08.870
And the mean is always
0, but the variance
00:13:08.870 --> 00:13:12.800
is determined by the
length of your interval.
00:13:12.800 --> 00:13:16.800
Exactly that will
be the variance.
00:13:16.800 --> 00:13:18.700
So try to remember
this property.
00:13:23.550 --> 00:13:28.375
A few more things, it has
a lot of different names.
00:13:28.375 --> 00:13:31.220
It's also called Wiener process.
00:13:36.710 --> 00:13:38.990
And let's see,
there was one more.
00:13:43.962 --> 00:13:45.170
Is there another name for it?
00:13:48.070 --> 00:13:52.100
I thought I had one more
name in mind, but maybe not.
00:13:52.100 --> 00:13:54.492
AUDIENCE: Norbert Wiener
was an MIT professor.
00:13:54.492 --> 00:13:55.325
PROFESSOR: Oh, yeah.
00:13:55.325 --> 00:13:56.440
That's important.
00:13:56.440 --> 00:13:58.200
AUDIENCE: Of course.
00:13:58.200 --> 00:14:02.420
PROFESSOR: Yeah, a
professor at MIT.
00:14:02.420 --> 00:14:04.650
But apparently he
wasn't the first person
00:14:04.650 --> 00:14:06.570
who discovered this process.
00:14:06.570 --> 00:14:10.450
I was some other person in 1900.
00:14:10.450 --> 00:14:13.260
And actually, in the
first paper that appeared,
00:14:13.260 --> 00:14:15.700
of course, they didn't know
about each other's result.
00:14:15.700 --> 00:14:17.200
In that paper the
reason he studied
00:14:17.200 --> 00:14:20.156
this was to evaluate stock
prices and auction prices.
00:14:36.620 --> 00:14:40.080
And here's another slightly
different description,
00:14:40.080 --> 00:14:41.740
maybe a more
intuitive description
00:14:41.740 --> 00:14:44.580
of the Brownian motion.
00:14:44.580 --> 00:14:46.720
So here is this philosophy.
00:14:46.720 --> 00:14:56.260
Philosophy is that Brownian
motion is the limit
00:14:56.260 --> 00:14:57.550
of simple random walks.
00:15:04.460 --> 00:15:08.429
The limit-- it's a
very vague concept.
00:15:08.429 --> 00:15:09.720
You'll see what I mean by this.
00:15:12.570 --> 00:15:18.210
So fix a time
interval of 0 up to 1
00:15:18.210 --> 00:15:21.470
and slice it into
very small pieces.
00:15:21.470 --> 00:15:23.480
So I'll say, into n pieces.
00:15:23.480 --> 00:15:27.290
1 over n, 2 over n, 3 over
n, dot dot dot, to n minus 1
00:15:27.290 --> 00:15:29.520
over n.
00:15:29.520 --> 00:15:31.400
And consider a
simple random walk,
00:15:31.400 --> 00:15:33.560
n-step simple random walk.
00:15:33.560 --> 00:15:35.860
So from time 0 you go
up or down, up or down.
00:15:39.290 --> 00:15:41.190
Then you get
something like that.
00:15:41.190 --> 00:15:43.940
OK?
00:15:43.940 --> 00:15:47.491
So let me be a little
bit more precise.
00:15:47.491 --> 00:16:01.380
Let Y_0, Y_1, to Y_n,
be a simple random walk,
00:16:01.380 --> 00:16:06.690
and let Z be the function
such that at time t over n,
00:16:06.690 --> 00:16:10.440
we let it to be Y of t.
00:16:10.440 --> 00:16:13.630
That's exactly just written
down in formula what it means.
00:16:13.630 --> 00:16:17.445
So this process is Z. I
take a simple random walk
00:16:17.445 --> 00:16:20.220
and scale it so that it
goes from time 0 to time 1.
00:16:24.180 --> 00:16:27.150
And then in the
intermediate values--
00:16:27.150 --> 00:16:29.040
for values that
are not this, just
00:16:29.040 --> 00:16:35.480
linearly extended-- linearly
extend in intermediate values.
00:16:41.570 --> 00:16:44.120
It's a complicated way of
saying just connect the dots.
00:16:49.390 --> 00:16:50.555
And take n to infinity.
00:16:55.870 --> 00:17:00.810
Then the resulting distribution
is a Brownian motion.
00:17:19.420 --> 00:17:21.660
So mathematically,
that's just saying
00:17:21.660 --> 00:17:24.369
the limit of simple random
walks is a Brownian motion.
00:17:24.369 --> 00:17:26.480
But it's more than that.
00:17:26.480 --> 00:17:29.430
That means if you
have some suspicion
00:17:29.430 --> 00:17:33.600
that some physical quantity
follows a Brownian motion,
00:17:33.600 --> 00:17:36.890
and then you
observe the variable
00:17:36.890 --> 00:17:41.640
at discrete times at
very, very fine scales--
00:17:41.640 --> 00:17:44.500
so you observe it really, really
often, like a million times
00:17:44.500 --> 00:17:46.640
in one second.
00:17:46.640 --> 00:17:51.877
Then once you see-- if you see
that and take it to the limit,
00:17:51.877 --> 00:17:53.210
it looks like a Brownian motion.
00:17:53.210 --> 00:17:56.140
Then now you can conclude
that it's a Brownian motion.
00:17:56.140 --> 00:18:01.300
What I'm trying to say is
this continuous time process,
00:18:01.300 --> 00:18:06.560
whatever the strange thing
is, it follows from something
00:18:06.560 --> 00:18:07.790
from a discrete world.
00:18:07.790 --> 00:18:10.200
It's not something new.
00:18:10.200 --> 00:18:13.350
It's the limit of these
objects that you already now.
00:18:17.440 --> 00:18:21.320
So this tells you that it might
be a reasonable model for stock
00:18:21.320 --> 00:18:23.380
prices because for
stock prices, no matter
00:18:23.380 --> 00:18:27.120
how-- there's only a
finite amount of time scale
00:18:27.120 --> 00:18:29.360
that you can observe the prices.
00:18:29.360 --> 00:18:32.220
But still, if you
observe it infinitely as
00:18:32.220 --> 00:18:34.750
much as you can, and
the distribution looks
00:18:34.750 --> 00:18:37.560
like a Brownian motion,
then you can use
00:18:37.560 --> 00:18:40.150
a Brownian motion to model it.
00:18:40.150 --> 00:18:43.860
So it's not only the
theoretical observation.
00:18:43.860 --> 00:18:46.930
It also has implication
when you want
00:18:46.930 --> 00:18:49.610
to use Brownian motion
as a physical model
00:18:49.610 --> 00:18:53.000
for some quantity.
00:18:53.000 --> 00:18:56.097
It also tells you why
Brownian motion might
00:18:56.097 --> 00:18:57.180
appear in some situations.
00:19:00.960 --> 00:19:02.155
So here's an example.
00:19:05.480 --> 00:19:07.420
Here's a completely
different context
00:19:07.420 --> 00:19:11.180
where Brownian motion
was discovered,
00:19:11.180 --> 00:19:14.220
and why it has the
name Brownian motion.
00:19:14.220 --> 00:19:18.960
So a botanist-- I don't know if
I'm pronouncing it correctly--
00:19:18.960 --> 00:19:26.210
named Brown in the
1800s, what he did was he
00:19:26.210 --> 00:19:39.630
observed a pollen
particle in water.
00:19:43.895 --> 00:19:46.020
So you have a cup of water
and there's some pollen.
00:19:48.660 --> 00:19:53.960
Of course you have gravity
that pulls the pollen down.
00:19:53.960 --> 00:19:56.690
And pollen is heavier than
water so eventually it
00:19:56.690 --> 00:19:59.460
will go down, eventually.
00:19:59.460 --> 00:20:01.360
But that only explains
the vertical action,
00:20:01.360 --> 00:20:02.976
it will only go down.
00:20:02.976 --> 00:20:04.850
But in fact, if you
observe what's happening,
00:20:04.850 --> 00:20:07.500
it just bounces back
and forth crazily
00:20:07.500 --> 00:20:12.100
until it finally reaches
down the bottom of your cup.
00:20:12.100 --> 00:20:15.460
And this motion,
if you just look
00:20:15.460 --> 00:20:17.760
at a two-dimension picture,
it's a Brownian motion
00:20:17.760 --> 00:20:18.810
to the left and right.
00:20:23.300 --> 00:20:29.522
So it moves as according
to Brownian motion.
00:20:35.370 --> 00:20:38.820
Well, first of all, I should
say a little bit more.
00:20:38.820 --> 00:20:40.550
What Brown did was
he observed it.
00:20:40.550 --> 00:20:44.595
He wasn't able to explain the
horizontal actions because he
00:20:44.595 --> 00:20:47.920
only understood
gravity, but then people
00:20:47.920 --> 00:20:49.120
tried to explain it.
00:20:49.120 --> 00:20:53.960
They suspected that it was
the water molecules that
00:20:53.960 --> 00:20:58.210
caused this action, but weren't
able to really explain it.
00:20:58.210 --> 00:21:01.100
But the first person to
actually rigorously explain it
00:21:01.100 --> 00:21:08.155
was, surprisingly,
Einstein, that relativity
00:21:08.155 --> 00:21:11.430
guy, that famous guy.
00:21:11.430 --> 00:21:13.680
So I was really surprised.
00:21:13.680 --> 00:21:16.595
He's really smart, apparently.
00:21:19.720 --> 00:21:21.870
And why?
00:21:21.870 --> 00:21:23.930
So why will this follow
a Brownian motion?
00:21:23.930 --> 00:21:25.760
Why is it a reasonable model?
00:21:25.760 --> 00:21:30.790
And this gives you a fairly
good reason for that.
00:21:30.790 --> 00:21:35.020
This description, where it's the
limit of simple random walks.
00:21:35.020 --> 00:21:37.030
Because if you think
about it, what's happening
00:21:37.030 --> 00:21:38.488
is there is a big
molecule that you
00:21:38.488 --> 00:21:42.050
can observe, this big particle.
00:21:42.050 --> 00:21:45.030
But inside there's
tiny water molecules,
00:21:45.030 --> 00:21:49.970
tiny ones that don't really
see, but it's filling the space.
00:21:49.970 --> 00:21:51.750
And they're just moving crazily.
00:21:51.750 --> 00:21:54.660
Even though the water looks
still, what's really happening
00:21:54.660 --> 00:21:56.990
is these water
molecules are just
00:21:56.990 --> 00:22:00.300
crazily moving inside the cup.
00:22:00.300 --> 00:22:07.170
And each water molecule, when
they collide with the pollen,
00:22:07.170 --> 00:22:10.510
it will change the action
of the pollen a little bit,
00:22:10.510 --> 00:22:13.170
by a tiny amount.
00:22:13.170 --> 00:22:18.770
So if you think about each
collision as one step,
00:22:18.770 --> 00:22:23.070
then each step will either
push this pollen to the left
00:22:23.070 --> 00:22:26.850
or to the right by
some tiny amount.
00:22:26.850 --> 00:22:28.810
And it just
accumulates over time.
00:22:28.810 --> 00:22:31.244
So you're looking at a
very, very fine time scale.
00:22:31.244 --> 00:22:33.160
Of course, the times
will differ a little bit,
00:22:33.160 --> 00:22:35.770
but let's just forget about
it, assume that it's uniform.
00:22:35.770 --> 00:22:38.360
And at each time it just
pushes to the left or right
00:22:38.360 --> 00:22:40.030
by a tiny amount.
00:22:40.030 --> 00:22:43.360
And you look at what
accumulates, as we saw,
00:22:43.360 --> 00:22:47.050
the limit of a simple random
walk is a Brownian motion.
00:22:47.050 --> 00:22:49.130
And that tells you why
we should get something
00:22:49.130 --> 00:22:50.550
like a Brownian motion here.
00:22:54.500 --> 00:23:03.515
So the action of pollen
particle is determined
00:23:03.515 --> 00:23:07.536
by infinitesimal-- I don't
know if that's the right word--
00:23:07.536 --> 00:23:18.250
but just, quote,
"infinitesimal" interactions
00:23:18.250 --> 00:23:19.310
with water molecules.
00:23:25.680 --> 00:23:29.670
That explains, at
least intuitively,
00:23:29.670 --> 00:23:31.430
why it follows Brownian motion.
00:23:35.610 --> 00:23:45.932
And the second example
is-- any questions here--
00:23:45.932 --> 00:23:47.045
is stock prices.
00:23:51.806 --> 00:23:55.670
At least to give you some
reasonable reason, some reason
00:23:55.670 --> 00:24:03.040
that Brownian motion is not so
bad a model for stock prices.
00:24:03.040 --> 00:24:11.980
Because if you look
at a stock price, S,
00:24:11.980 --> 00:24:14.700
the price is determined by
buying actions or selling
00:24:14.700 --> 00:24:16.100
actions.
00:24:16.100 --> 00:24:18.220
Each action kind of
pulls down the price
00:24:18.220 --> 00:24:20.180
or pulls up the price,
pushes down the price
00:24:20.180 --> 00:24:21.660
or pulls up the price.
00:24:21.660 --> 00:24:26.110
And if you look at very, very
tiny scales, what's happening
00:24:26.110 --> 00:24:29.690
is at a very tiny amount
they will go up or down.
00:24:29.690 --> 00:24:32.230
Of course, it doesn't go up
and down by a uniform amount,
00:24:32.230 --> 00:24:34.640
but just forget about
that technicality.
00:24:34.640 --> 00:24:37.640
It just bounces back and
forth infinitely often,
00:24:37.640 --> 00:24:40.045
and then you're taking
these tiny scales
00:24:40.045 --> 00:24:42.590
to be tinier, so
very, very small.
00:24:42.590 --> 00:24:45.185
So again, you see
this limiting picture.
00:24:45.185 --> 00:24:47.060
Where you have a discrete--
something looking
00:24:47.060 --> 00:24:51.540
like a random walk, and
you take t as infinity.
00:24:51.540 --> 00:24:55.830
So if that's the only
action causing the price,
00:24:55.830 --> 00:25:00.020
then Brownian motion will
be the right model to use.
00:25:00.020 --> 00:25:02.930
Of course, there are many
other things involved
00:25:02.930 --> 00:25:06.900
which makes this deviate
from Brownian motion,
00:25:06.900 --> 00:25:09.850
but at least, theoretically,
it's a good starting point.
00:25:15.270 --> 00:25:16.155
Any questions?
00:25:19.420 --> 00:25:20.020
OK.
00:25:20.020 --> 00:25:21.145
So you saw Brownian motion.
00:25:21.145 --> 00:25:23.750
You already know that it's used
in the financial market a lot.
00:25:23.750 --> 00:25:27.130
It's also being used in science
and other fields like that.
00:25:27.130 --> 00:25:31.562
And really big names, like
Einstein, is involved.
00:25:31.562 --> 00:25:33.770
So it's a really, really
important theoretical thing.
00:25:37.750 --> 00:25:41.270
Now that you've learned it,
it's time to get used to it.
00:25:43.890 --> 00:25:45.600
So I'll tell you
some properties,
00:25:45.600 --> 00:25:48.620
and actually prove a little
bit-- just some propositions
00:25:48.620 --> 00:25:50.650
to show you some properties.
00:25:50.650 --> 00:25:53.970
Some of them are quite
surprising if you never
00:25:53.970 --> 00:25:56.160
saw it before.
00:25:56.160 --> 00:25:56.660
OK.
00:25:56.660 --> 00:25:57.826
So here are some properties.
00:26:05.990 --> 00:26:17.146
Crosses the x-axis
infinitely often,
00:26:17.146 --> 00:26:18.270
or I should say the t-axis.
00:26:21.390 --> 00:26:25.375
Because you start from 0, it
will never go to infinity,
00:26:25.375 --> 00:26:27.230
or get to negative infinity.
00:26:27.230 --> 00:26:29.330
It will always go balanced
positive and negative
00:26:29.330 --> 00:26:31.496
infinitely often.
00:26:31.496 --> 00:26:42.636
And the second, it does
not deviate too much
00:26:42.636 --> 00:26:48.748
from t equals y squared.
00:26:48.748 --> 00:26:51.380
We'll call this y.
00:26:51.380 --> 00:26:52.880
Now, this is a very
vague statement.
00:26:52.880 --> 00:26:57.554
What I'm trying to say is
draw this curve as this.
00:27:03.530 --> 00:27:08.430
If you start at time
0, at some time t_0,
00:27:08.430 --> 00:27:10.120
the probability
distribution here
00:27:10.120 --> 00:27:11.860
is given as a normal
random variable
00:27:11.860 --> 00:27:16.400
with mean 0 and variance t_0.
00:27:16.400 --> 00:27:21.309
And because of that,
the standard deviation
00:27:21.309 --> 00:27:22.100
is square root t_0.
00:27:26.740 --> 00:27:30.960
So the typical value will be
around the standard deviation.
00:27:30.960 --> 00:27:32.030
And it won't deviate.
00:27:32.030 --> 00:27:33.720
It can be 100 times this.
00:27:33.720 --> 00:27:37.370
It won't really be a million
times that or something.
00:27:37.370 --> 00:27:42.475
So most likely it will
look something like that.
00:27:45.325 --> 00:27:48.170
So it plays around
this curve a lot,
00:27:48.170 --> 00:27:50.195
but it crosses the
axis infinitely often.
00:27:50.195 --> 00:27:52.962
It goes back and forth.
00:27:52.962 --> 00:27:53.830
What else?
00:27:53.830 --> 00:27:56.710
The third one is quite
really interesting.
00:27:56.710 --> 00:27:59.010
It's more theoretical
interest, but it also
00:27:59.010 --> 00:28:01.510
has real-life implications.
00:28:01.510 --> 00:28:12.140
It's not differentiable
anywhere.
00:28:12.140 --> 00:28:15.500
It's nowhere differentiable.
00:28:15.500 --> 00:28:18.140
So this curve,
whatever that curve is,
00:28:18.140 --> 00:28:21.691
it's a continuous path, but it's
nowhere differentiable, really
00:28:21.691 --> 00:28:22.190
surprising.
00:28:22.190 --> 00:28:24.970
It's hard to imagine
even one such path.
00:28:24.970 --> 00:28:27.640
What it's saying is if you
take one path according
00:28:27.640 --> 00:28:30.270
to this probability
distribution,
00:28:30.270 --> 00:28:32.580
then more than likely
you'll obtain a path which
00:28:32.580 --> 00:28:33.663
is nowhere differentiable.
00:28:36.560 --> 00:28:40.972
That just sounds nice,
but why it does it matter?
00:28:40.972 --> 00:28:44.860
It matters because we
can't use calculus anymore.
00:28:53.680 --> 00:28:55.180
Because all the
theory of calculus
00:28:55.180 --> 00:28:58.360
is based on differentiation.
00:28:58.360 --> 00:29:02.210
However, our paths have some
nice things, it's universal,
00:29:02.210 --> 00:29:05.190
and it appears in very
different contexts.
00:29:05.190 --> 00:29:07.600
But if you want to
do analysis on it,
00:29:07.600 --> 00:29:09.500
it's just not differentiable.
00:29:09.500 --> 00:29:12.020
So the standard
tools of calculus
00:29:12.020 --> 00:29:15.270
can't be used here, which
is quite unfortunate
00:29:15.270 --> 00:29:16.340
if you think about it.
00:29:16.340 --> 00:29:19.860
You have this nice model,
which can describe many things,
00:29:19.860 --> 00:29:21.710
you can't really
do analysis on it.
00:29:25.052 --> 00:29:26.510
We'll later see
that actually there
00:29:26.510 --> 00:29:34.780
is a variant, a different
calculus that works.
00:29:37.640 --> 00:29:40.920
And I'm sure many of you
would have heard about it.
00:29:40.920 --> 00:29:42.100
It's called Ito's calculus.
00:29:48.810 --> 00:29:50.210
So we have this nice object.
00:29:50.210 --> 00:29:52.040
Unfortunately, it's
not differentiable,
00:29:52.040 --> 00:29:54.710
so the standard calculus
does not work here.
00:29:54.710 --> 00:29:57.640
However, there is
a modified version
00:29:57.640 --> 00:30:02.110
of calculus called
Ito's calculus, which
00:30:02.110 --> 00:30:04.310
extends the classical
calculus to this setting.
00:30:04.310 --> 00:30:06.930
And it's really powerful
and it's really cool.
00:30:06.930 --> 00:30:10.310
But unfortunately, we don't
have that much time to cover it.
00:30:10.310 --> 00:30:13.920
I will only be able to tell
you really basic properties
00:30:13.920 --> 00:30:17.270
and basic computations of it.
00:30:17.270 --> 00:30:22.150
And you'll see how
this calculus is
00:30:22.150 --> 00:30:24.730
being used in the
financial world
00:30:24.730 --> 00:30:26.330
in the coming-up lectures.
00:30:33.080 --> 00:30:34.950
But before going
into Ito's calculus,
00:30:34.950 --> 00:30:38.220
let's talk about the property
of Brownian motion a little bit
00:30:38.220 --> 00:30:40.316
because we have
to get used to it.
00:30:47.760 --> 00:30:52.960
Suppose I'm using it as
a model of a stock price.
00:30:52.960 --> 00:31:00.190
So I'm using-- use
Brownian motion
00:31:00.190 --> 00:31:12.390
as a model for stock price--
say, daily stock price.
00:31:16.730 --> 00:31:21.570
The market opens at 9:30 AM.
00:31:21.570 --> 00:31:24.890
It closes at 4:00 PM.
00:31:24.890 --> 00:31:31.228
It starts at some
price, and then moves
00:31:31.228 --> 00:31:32.602
according to the
Brownian motion.
00:31:37.690 --> 00:31:43.370
And then you want to obtain the
distribution of the min value
00:31:43.370 --> 00:31:45.340
and the max value for the stock.
00:31:49.510 --> 00:31:52.720
So these are very
useful statistics.
00:31:52.720 --> 00:31:56.080
So a daily stock
price, what will
00:31:56.080 --> 00:31:58.610
the minimum and the
maximum-- what will
00:31:58.610 --> 00:32:01.490
the distribution of those be?
00:32:01.490 --> 00:32:02.380
So let's compute it.
00:32:02.380 --> 00:32:03.570
We can actually compute it.
00:32:10.920 --> 00:32:14.670
What we want to do is-- I'll
just compute the maximum.
00:32:14.670 --> 00:32:20.166
I want to compute this
thing over s smaller
00:32:20.166 --> 00:32:23.415
than t of the Brownian motion.
00:32:28.250 --> 00:32:33.650
So I define this new process
from the Brownian motion,
00:32:33.650 --> 00:32:35.630
and I want to compute
the distribution
00:32:35.630 --> 00:32:39.830
of this new stochastic process.
00:32:39.830 --> 00:32:40.930
And here's the theorem.
00:32:44.300 --> 00:32:51.160
So for all t, the
probability that you
00:32:51.160 --> 00:33:04.410
have M(t) greater than a and
positive a is equal to 2 times
00:33:04.410 --> 00:33:11.620
the probability that you have
the Brownian motion greater
00:33:11.620 --> 00:33:12.120
than a.
00:33:17.274 --> 00:33:18.190
It's quite surprising.
00:33:20.710 --> 00:33:22.650
If you just look
at this, there's
00:33:22.650 --> 00:33:26.290
no reason to expect that
such a nice formula should
00:33:26.290 --> 00:33:27.050
exist at all.
00:33:31.340 --> 00:33:34.640
And notice that maximum
is always at least 0,
00:33:34.640 --> 00:33:37.390
so we don't have to worry
about negative values.
00:33:37.390 --> 00:33:38.585
It starts at 0.
00:33:41.100 --> 00:33:42.295
How do we prove it?
00:33:48.328 --> 00:33:48.828
Proof.
00:33:52.650 --> 00:33:53.820
Take this tau.
00:33:53.820 --> 00:33:57.350
It's a stopping time, if
you remember what it is.
00:33:57.350 --> 00:34:09.389
It's a minimum value of t
such that the Brownian motion
00:34:09.389 --> 00:34:10.630
at time t is equal to a.
00:34:13.594 --> 00:34:15.260
That's a complicated
way of saying, just
00:34:15.260 --> 00:34:17.587
record the first time
you hit the line a.
00:34:21.403 --> 00:34:24.760
Line a, with some
Brownian motion,
00:34:24.760 --> 00:34:26.780
and you record this time.
00:34:26.780 --> 00:34:28.380
That will be your tau of a.
00:34:56.389 --> 00:35:00.610
So now here's some
strange thing.
00:35:00.610 --> 00:35:54.530
The probability that B(t),
B(tau_a), given this-- OK.
00:35:54.530 --> 00:36:01.850
So what this is saying is, if
you're interested at time t,
00:36:01.850 --> 00:36:05.080
if your tau_a happened
before time t,
00:36:05.080 --> 00:36:10.100
so if your Brownian motion
hit the line a before time t,
00:36:10.100 --> 00:36:14.280
then afterwards you have the
same probability of ending up
00:36:14.280 --> 00:36:18.350
above a and ending up below a.
00:36:18.350 --> 00:36:21.150
The reason is because you
can just reflect the path.
00:36:21.150 --> 00:36:24.220
Whatever path that
ends over a, you
00:36:24.220 --> 00:36:29.450
can reflect it to obtain
a path that ends below a.
00:36:29.450 --> 00:36:31.475
And by symmetry, you
just have this property.
00:36:34.520 --> 00:36:36.888
Well, it's not obvious how
you'll use this right now.
00:36:49.110 --> 00:36:51.540
And then we're almost done.
00:36:51.540 --> 00:36:56.840
The probability that maximum
at time t is greater than a
00:36:56.840 --> 00:37:00.710
that's equal to the probability
that you're stopping time
00:37:00.710 --> 00:37:02.985
is less than t,
just by definition.
00:37:06.510 --> 00:37:12.425
And that's equal to the
probability that B(t) minus
00:37:12.425 --> 00:37:18.940
B(tau_a) is positive given
tau a is less than t--
00:37:35.840 --> 00:37:39.350
Because if you know
that tau is less than t,
00:37:39.350 --> 00:37:41.270
there's only two possible ways.
00:37:41.270 --> 00:37:44.780
You can either go up afterwards,
or you can go down afterwards.
00:37:44.780 --> 00:37:47.410
But these two are
the same probability.
00:37:47.410 --> 00:37:59.070
What you obtain is 2 times the
probability that-- and that's
00:37:59.070 --> 00:38:00.920
just equal to 2
times the probability
00:38:00.920 --> 00:38:03.620
that B(t) is greater than a.
00:38:14.790 --> 00:38:15.500
What happened?
00:38:15.500 --> 00:38:16.630
Some magic happened.
00:38:16.630 --> 00:38:18.463
First of all, these two
are the same because
00:38:18.463 --> 00:38:20.230
of this property by symmetry.
00:38:20.230 --> 00:38:25.560
Then from here to here, B(tau_a)
is always equal to a, as long
00:38:25.560 --> 00:38:27.230
as tau_a is less than t.
00:38:27.230 --> 00:38:32.110
This is just-- I rewrote this
as a, and I got this thing.
00:38:32.110 --> 00:38:35.980
And then I can just remove
this because if I already
00:38:35.980 --> 00:38:42.480
know that tau_a is less
than t-- order is reversed.
00:38:42.480 --> 00:38:45.850
If I already know that B at
time t is greater than a,
00:38:45.850 --> 00:38:47.460
then I know that
tau is less than t.
00:38:47.460 --> 00:38:51.780
Because if you want to reach
a because of continuity,
00:38:51.780 --> 00:38:55.340
if you want to go over a, you
have to reach a at some point.
00:38:55.340 --> 00:38:58.790
That means you hit
a before time t.
00:38:58.790 --> 00:39:03.320
So that event is already
inside that event.
00:39:03.320 --> 00:39:05.350
And you just get rid of it.
00:39:23.542 --> 00:39:27.740
Sorry, all this should
be-- something looks weird.
00:39:31.853 --> 00:39:32.660
Not conditioned.
00:39:39.980 --> 00:39:40.480
OK.
00:39:40.480 --> 00:39:41.476
That makes more sense.
00:39:46.589 --> 00:39:48.255
Just the intersection
of two properties.
00:39:52.850 --> 00:39:54.140
Any questions here?
00:40:01.190 --> 00:40:04.890
So again, you just want
to compute the probability
00:40:04.890 --> 00:40:08.760
that the maximum is
greater than a at time t.
00:40:11.460 --> 00:40:14.050
In other words, just
by definition of tau_a,
00:40:14.050 --> 00:40:18.310
that's equal to the problem
that tau_a is less than t.
00:40:18.310 --> 00:40:20.940
And if tau_a is less
than t, afterwards,
00:40:20.940 --> 00:40:23.470
depending on afterwards
what happens,
00:40:23.470 --> 00:40:24.960
it increases or decreases.
00:40:24.960 --> 00:40:26.450
So there's only
two possibilities.
00:40:26.450 --> 00:40:29.490
It increases or it decreases.
00:40:29.490 --> 00:40:31.470
But these two events
have the same probability
00:40:31.470 --> 00:40:32.511
because of this property.
00:40:35.490 --> 00:40:38.180
Here's a bar and
that's an intersection.
00:40:38.180 --> 00:40:43.430
But it doesn't matter, because
if you have the B of X_1 bar y
00:40:43.430 --> 00:40:48.000
equals B of x_2 bar
y then probability
00:40:48.000 --> 00:40:51.580
of X_1 intersection Y
over probability of Y
00:40:51.580 --> 00:41:00.110
is equal to-- these two cancel.
00:41:00.110 --> 00:41:05.400
So this bar can just be
replaced by intersection.
00:41:05.400 --> 00:41:07.920
That means these two events
have the same probability.
00:41:07.920 --> 00:41:09.140
So you can just take one.
00:41:09.140 --> 00:41:12.040
What I'm going to take
is one that goes above 0.
00:41:12.040 --> 00:41:16.007
So after tau_a, it
accumulates more value.
00:41:16.007 --> 00:41:18.090
And if you rewrite it,
what that means is just B_t
00:41:18.090 --> 00:41:21.640
is greater than a given
that tau_a is less than t.
00:41:21.640 --> 00:41:24.920
But now that just
became redundant.
00:41:24.920 --> 00:41:27.440
Because if you already know
that B(t) is greater than a,
00:41:27.440 --> 00:41:30.380
tau_a has to be less than t.
00:41:30.380 --> 00:41:31.880
And that's just the conclusion.
00:41:35.696 --> 00:41:39.910
And it's just some nice
result about the maximum
00:41:39.910 --> 00:41:41.065
over some time interval.
00:41:44.660 --> 00:41:51.425
And actually, I think Peter uses
distribution in your lecture,
00:41:51.425 --> 00:41:51.925
right?
00:41:51.925 --> 00:41:53.656
AUDIENCE: Yes.
00:41:53.656 --> 00:42:00.052
[INAUDIBLE] is that the
distribution of the max
00:42:00.052 --> 00:42:03.988
minus the movement of
the Brownian motion.
00:42:03.988 --> 00:42:07.520
And use that range of
the process as a scaling
00:42:07.520 --> 00:42:13.942
for [INAUDIBLE] and get more
precise measures of volatility
00:42:13.942 --> 00:42:17.888
than just using, say,
the close-to-close price
00:42:17.888 --> 00:42:18.388
[INAUDIBLE].
00:42:23.064 --> 00:42:23.730
PROFESSOR: Yeah.
00:42:32.250 --> 00:42:34.301
That was one property.
00:42:34.301 --> 00:43:15.290
And another property is-- and
that's what I already told you,
00:43:15.290 --> 00:43:16.570
but I'm going to prove this.
00:43:16.570 --> 00:43:18.904
So at each time
the Brownian motion
00:43:18.904 --> 00:43:20.320
is not differentiable
at that time
00:43:20.320 --> 00:43:23.910
with probability equal to 1.
00:43:23.910 --> 00:43:26.520
Well, not very
strictly, but I will
00:43:26.520 --> 00:43:28.830
use this theorem to prove it.
00:43:28.830 --> 00:43:29.330
OK?
00:43:41.940 --> 00:43:49.260
Suppose the Brownian motion
has a differentiation at time t
00:43:49.260 --> 00:43:50.660
and it's equal to a.
00:43:59.114 --> 00:44:01.530
Then what you just see is that
the Brownian motion at time
00:44:01.530 --> 00:44:10.480
t plus epsilon, minus
Brownian motion at time t,
00:44:10.480 --> 00:44:15.260
has to be less than or
equal to epsilon times a.
00:44:15.260 --> 00:44:19.435
Not precisely, so
I'll say just almost.
00:44:23.011 --> 00:44:24.510
Can make it
mathematically rigorous.
00:44:24.510 --> 00:44:26.410
But what I'm trying
to say here is
00:44:26.410 --> 00:44:29.250
by-- is it mean value theorem?
00:44:29.250 --> 00:44:36.430
So from t to t plus epsilon, you
expect to gain a times epsilon.
00:44:36.430 --> 00:44:40.540
That's-- OK?
00:44:40.540 --> 00:44:43.160
You should have this-- then.
00:44:45.770 --> 00:44:46.960
In fact, for all epsilon.
00:44:53.690 --> 00:44:57.141
Greater than epsilon prime'.
00:44:57.141 --> 00:44:59.250
Let's write it like that.
00:44:59.250 --> 00:45:04.960
So in other words, the
maximum in this interval,
00:45:04.960 --> 00:45:07.470
B(t+epsilon) minus t, this
distribution is the same
00:45:07.470 --> 00:45:09.880
as the maximum at epsilon prime.
00:45:09.880 --> 00:45:14.640
That has to be less
than epsilon times A. So
00:45:14.640 --> 00:45:18.670
what I'm trying to say is if
this differentiable, depending
00:45:18.670 --> 00:45:23.430
on the slope, your Brownian
motion should have always been
00:45:23.430 --> 00:45:29.160
inside this cone from t
up to time t plus epsilon.
00:45:29.160 --> 00:45:34.150
If you draw this slope, it must
have been inside this cone.
00:45:34.150 --> 00:45:38.180
I'm trying to say that
this cannot happen.
00:45:38.180 --> 00:45:40.340
From here to here, it
should have passed this line
00:45:40.340 --> 00:45:41.640
at some point.
00:45:41.640 --> 00:45:42.345
OK?
00:45:42.345 --> 00:45:44.220
So to do that I'm looking
at the distribution
00:45:44.220 --> 00:45:47.560
of the maximum value
over this time interval.
00:45:47.560 --> 00:45:50.066
And I want to say that it's
even greater than that.
00:45:50.066 --> 00:45:52.460
So if your maximum
is greater than that,
00:45:52.460 --> 00:45:55.520
you definitely can't
have this control.
00:45:55.520 --> 00:46:04.650
So if differentiable,
then maximum of epsilon
00:46:04.650 --> 00:46:17.595
prime-- the maximum of epsilon,
actually, and just compute it.
00:46:17.595 --> 00:46:23.520
So the probability that M
epsilon is less than epsilon*A
00:46:23.520 --> 00:46:27.390
is equal to 2 times the
probability of that,
00:46:27.390 --> 00:46:33.297
the Brownian motion at epsilon
is less than or equal to a.
00:46:33.297 --> 00:46:34.505
This has normal distribution.
00:46:39.750 --> 00:46:44.440
And if you normalize
it to N(0, 1),
00:46:44.440 --> 00:46:47.691
divide by the standard deviation
so you get the square root
00:46:47.691 --> 00:46:51.390
of epsilon A.
00:46:51.390 --> 00:46:55.227
As epsilon goes to
0, this goes to 0.
00:46:55.227 --> 00:46:56.435
That means this goes to half.
00:46:59.037 --> 00:47:00.120
The whole thing goes to 1.
00:47:05.760 --> 00:47:06.624
What am I missing?
00:47:06.624 --> 00:47:07.540
I did something wrong.
00:47:07.540 --> 00:47:10.744
I flipped it.
00:47:10.744 --> 00:47:11.410
This is greater.
00:47:18.590 --> 00:47:20.880
Now, if you combine it,
if it was differentiable,
00:47:20.880 --> 00:47:24.120
your maximum should have
been less than epsilon*A.
00:47:24.120 --> 00:47:26.860
But what we saw here is your
maximum is always greater than
00:47:26.860 --> 00:47:29.770
that epsilon times A.
With probability 1,
00:47:29.770 --> 00:47:31.200
you take epsilon goes to 0.
00:47:40.980 --> 00:47:41.958
Any questions?
00:47:46.380 --> 00:47:48.530
OK.
00:47:48.530 --> 00:47:50.750
So those are some
interesting things,
00:47:50.750 --> 00:47:53.280
properties of Brownian motion
that I want to talk about.
00:47:53.280 --> 00:47:56.510
I have one final thing,
and this one it's
00:47:56.510 --> 00:48:00.420
really important theoretically.
00:48:00.420 --> 00:48:07.170
And also, it will be the main
lemma for Ito's calculus.
00:48:07.170 --> 00:48:11.090
So the theorem is called
quadratic variation.
00:48:19.507 --> 00:48:21.590
And it's something that
doesn't happen that often.
00:48:29.860 --> 00:49:14.050
So let 0-- let me write
it down even more clear.
00:49:39.894 --> 00:49:42.390
Now that's something strange.
00:49:42.390 --> 00:49:47.220
Let me just first parse
it before proving it.
00:49:47.220 --> 00:49:50.084
Think about it as just
a function, function f.
00:49:54.730 --> 00:49:55.720
What is this quantity?
00:49:55.720 --> 00:49:59.141
This quantity means that
from 0 up to time T,
00:49:59.141 --> 00:50:03.240
you chop it up into n pieces.
00:50:03.240 --> 00:50:07.810
You get T over n, 2T
over n, 3T over n,
00:50:07.810 --> 00:50:10.570
and you look at the function.
00:50:10.570 --> 00:50:15.350
The difference between
each consecutive points,
00:50:15.350 --> 00:50:19.340
record these differences,
and then square it.
00:50:19.340 --> 00:50:21.770
And you sum it as
n goes to infinity.
00:50:21.770 --> 00:50:26.740
So you take smaller and smaller
scales take it to infinity.
00:50:26.740 --> 00:50:28.840
What the theorem says
is for Brownian motion
00:50:28.840 --> 00:50:31.560
this goes to T, the limit.
00:50:31.560 --> 00:50:32.875
Why is this something strange?
00:50:42.640 --> 00:50:44.500
Assume f is a lot
better function.
00:50:44.500 --> 00:50:49.220
Assume f is continuously
differentiable.
00:50:49.220 --> 00:50:52.880
That means it's differentiable,
and its differentiation
00:50:52.880 --> 00:50:54.880
is continuous.
00:50:54.880 --> 00:50:56.109
Derivative is continuous.
00:51:01.600 --> 00:51:04.420
Then let's compute the
exact same property,
00:51:04.420 --> 00:51:05.260
exact same thing.
00:51:05.260 --> 00:51:08.770
I'll just call this--
maybe i will be better.
00:51:11.510 --> 00:51:20.490
This time t_i and time t_(i-1),
then the sum over i of f
00:51:20.490 --> 00:51:24.446
at t_(i+1) minus f at t_i.
00:51:24.446 --> 00:51:31.330
If you square it, this is at
most sum from i equal 1 to n,
00:51:31.330 --> 00:51:42.190
f of t_(i+1) minus f of t_i,
times-- by mean value theorem--
00:51:42.190 --> 00:51:43.620
f prime of s_i.
00:52:02.810 --> 00:52:06.340
So by mean value theorem, there
exists a point s_i such that
00:52:06.340 --> 00:52:09.740
f(t_(i+1)) minus f(t_i) is equal
to f prime s_i, times that.
00:52:09.740 --> 00:52:11.300
s_i belongs to that interval.
00:52:21.020 --> 00:52:22.500
Yes.
00:52:22.500 --> 00:52:24.660
And then you take this term out.
00:52:24.660 --> 00:52:32.850
You take the maximum, from 0
up to t, f prime of s squared,
00:52:32.850 --> 00:52:39.930
times i equal 1 to n,
t_(i+1) minus t_i squared.
00:52:39.930 --> 00:52:43.710
This thing is T over n
because we chopped it up
00:52:43.710 --> 00:52:44.950
into n intervals.
00:52:44.950 --> 00:52:47.220
Each consecutive
difference is T over n.
00:52:47.220 --> 00:52:51.480
If you square it, that's equal
to T squared over n squared.
00:52:51.480 --> 00:52:55.310
If you had n of them,
you get T squared over n.
00:52:55.310 --> 00:53:01.320
So you get whatever that maximum
is times T squared over n.
00:53:01.320 --> 00:53:03.430
If you take n to
infinity, that goes to 0.
00:53:06.170 --> 00:53:08.160
So if you have a
reasonable function, which
00:53:08.160 --> 00:53:11.290
is differentiable,
this variation--
00:53:11.290 --> 00:53:14.930
this is called the quadratic
variation-- quadratic variation
00:53:14.930 --> 00:53:16.320
is 0.
00:53:16.320 --> 00:53:19.800
So all these classical functions
that you've been studying
00:53:19.800 --> 00:53:22.610
will not even have this
quadratic variation.
00:53:22.610 --> 00:53:24.800
But for Brownian
motion, what's happening
00:53:24.800 --> 00:53:28.010
is it just bounced back
and forth too much.
00:53:28.010 --> 00:53:30.270
Even if you scale it
smaller and smaller,
00:53:30.270 --> 00:53:32.900
the variation is big
enough to accumulate.
00:53:32.900 --> 00:53:36.460
They won't disappear like if it
was a differentiable function.
00:53:39.110 --> 00:53:42.550
And that pretty much-- it's
a slightly stronger version
00:53:42.550 --> 00:53:44.175
than this that it's
not differentiable.
00:53:47.290 --> 00:53:49.354
We saw that it's
not differentiable.
00:53:49.354 --> 00:53:50.770
And this a different
way of saying
00:53:50.770 --> 00:53:51.978
that it's not differentiable.
00:53:54.596 --> 00:53:56.310
It has very important
implications.
00:54:00.530 --> 00:54:06.246
And another way to write it is--
so here's a difference of B,
00:54:06.246 --> 00:54:10.060
it's dB squared is equal to dt.
00:54:13.340 --> 00:54:15.090
So if you take the
differential-- whatever
00:54:15.090 --> 00:54:17.650
that means-- if you take
the infinitesimal difference
00:54:17.650 --> 00:54:21.690
of each side, this part
is just dB squared,
00:54:21.690 --> 00:54:27.130
the Brownian motion difference
squared; this part is d of t.
00:54:27.130 --> 00:54:31.662
And that we'll see again.
00:54:31.662 --> 00:54:33.620
But before that, let's
just prove this theorem.
00:54:57.730 --> 00:55:04.360
So we're looking at the sum of
B of t_(i+1), minus B of t_i,
00:55:04.360 --> 00:55:06.100
squared.
00:55:06.100 --> 00:55:10.140
Where t of i is i
over n times the time.
00:55:14.496 --> 00:55:17.958
From 1 to n-- 0 to n minus 1.
00:55:22.248 --> 00:55:22.748
OK.
00:55:26.647 --> 00:55:27.980
What's the distribution of this?
00:55:30.926 --> 00:55:32.400
AUDIENCE: Normal.
00:55:32.400 --> 00:55:40.310
PROFESSOR: Normal, meaning 0,
variance t_(i+1) minus t_i.
00:55:40.310 --> 00:55:41.455
But that was just T over n.
00:55:44.020 --> 00:55:46.720
Is the distribution.
00:55:46.720 --> 00:55:48.290
So I'll write it like this.
00:55:48.290 --> 00:55:51.666
You sum from i equal
1 to n minus 1,
00:55:51.666 --> 00:55:55.770
X_i squared for X_i
is normal variable.
00:56:07.306 --> 00:56:07.806
OK?
00:56:13.190 --> 00:56:15.550
And what's the expectation
of X_i squared?
00:56:19.674 --> 00:56:21.500
It's T squared over n squared.
00:56:27.590 --> 00:56:28.460
OK.
00:56:28.460 --> 00:56:31.034
So maybe it's better
to write it like this.
00:56:31.034 --> 00:56:34.940
So I'll just write it again--
the sum from i equals 0 to n
00:56:34.940 --> 00:56:39.720
minus 1 of random variables Y_i,
such that expectation of Y_i--
00:56:42.748 --> 00:56:43.664
AUDIENCE: [INAUDIBLE].
00:56:48.935 --> 00:56:52.040
PROFESSOR: Did I make
a mistake somewhere?
00:56:52.040 --> 00:56:55.549
AUDIENCE: The expected value
of X_i squared is the variance.
00:56:55.549 --> 00:56:56.590
PROFESSOR: It's T over n.
00:56:56.590 --> 00:56:57.920
Oh, yeah, you're right.
00:57:00.920 --> 00:57:01.530
Thank you.
00:57:10.590 --> 00:57:12.230
OK.
00:57:12.230 --> 00:57:17.666
So divide by n
and multiply by n.
00:57:21.240 --> 00:57:21.820
What is this?
00:57:21.820 --> 00:57:23.217
What will this go to?
00:57:33.654 --> 00:57:36.139
AUDIENCE: [INAUDIBLE].
00:57:36.139 --> 00:57:38.140
PROFESSOR: No.
00:57:38.140 --> 00:57:39.790
Remember strong law
of large numbers.
00:57:42.539 --> 00:57:44.080
You have a bunch of
random variables,
00:57:44.080 --> 00:57:45.496
which are independent,
identically
00:57:45.496 --> 00:57:48.896
distributed, and mean T over n.
00:57:48.896 --> 00:57:52.960
You sum n of them
and divide by n.
00:57:52.960 --> 00:57:55.860
You know that it just
converges to T over
00:57:55.860 --> 00:57:58.240
n, just this one number.
00:57:58.240 --> 00:58:02.240
It doesn't-- it's
a distribution,
00:58:02.240 --> 00:58:05.630
but most of the time
it's just T over n.
00:58:05.630 --> 00:58:06.590
OK?
00:58:06.590 --> 00:58:12.540
If you take-- that's
equal to T, because these
00:58:12.540 --> 00:58:14.180
are random variables
accumulating
00:58:14.180 --> 00:58:16.210
these squared terms.
00:58:16.210 --> 00:58:17.200
That's what's happened.
00:58:17.200 --> 00:58:21.880
Just a nice application of
strong law of large numbers,
00:58:21.880 --> 00:58:24.080
or just law of large numbers.
00:58:24.080 --> 00:58:25.574
To be precise,
you'll have to use
00:58:25.574 --> 00:58:26.740
strong law of large numbers.
00:58:42.806 --> 00:58:43.306
OK.
00:58:46.290 --> 00:58:48.650
So I think that's enough
for Brownian motion.
00:58:54.669 --> 00:58:55.460
And final question?
00:58:58.418 --> 00:58:58.918
OK.
00:59:02.880 --> 00:59:03.920
Now, let's move on--
00:59:03.920 --> 00:59:05.086
AUDIENCE: I have a question.
00:59:05.086 --> 00:59:05.808
PROFESSOR: Yes.
00:59:05.808 --> 00:59:10.056
AUDIENCE: So this
[INAUDIBLE], is it
00:59:10.056 --> 00:59:12.667
for all Brownian motions B?
00:59:12.667 --> 00:59:13.500
PROFESSOR: Oh, yeah.
00:59:13.500 --> 00:59:15.460
That's a good question.
00:59:15.460 --> 00:59:17.252
This is what happens
with probability one.
00:59:17.252 --> 00:59:20.282
So always-- I'll
just say always.
00:59:20.282 --> 00:59:21.490
It's not a very strict sense.
00:59:21.490 --> 00:59:24.180
But if you take one path
according to the Brownian
00:59:24.180 --> 00:59:28.950
motion, in that path
you'll have this.
00:59:28.950 --> 00:59:31.730
No matter what path you
get, it always happens.
00:59:34.496 --> 00:59:35.817
AUDIENCE: With probability one.
00:59:35.817 --> 00:59:37.150
PROFESSOR: With probability one.
00:59:37.150 --> 00:59:41.516
So there's a hiding
statement-- with probability.
00:59:45.970 --> 00:59:49.500
And you'll see why you need
this with probability one
00:59:49.500 --> 00:59:52.776
is because we're using this
probability statement here.
00:59:56.060 --> 01:00:00.465
But for all practical means,
like with probability one
01:00:00.465 --> 01:00:01.260
just means always.
01:00:11.000 --> 01:00:13.520
Now, I want to motivate
Ito's calculus.
01:00:19.760 --> 01:00:21.330
First of all, this.
01:00:24.620 --> 01:00:28.080
So now, I was saying that
Brownian motion, at least,
01:00:28.080 --> 01:00:32.660
is not so bad a model
for stock prices.
01:00:32.660 --> 01:00:35.080
But if you remember
what I said before,
01:00:35.080 --> 01:00:37.190
and what people
are actually doing,
01:00:37.190 --> 01:00:39.380
a better way to
describe it is instead
01:00:39.380 --> 01:00:42.970
of the differences being a
normal distribution, what
01:00:42.970 --> 01:00:45.580
we want is the
percentile difference.
01:00:45.580 --> 01:01:02.190
So for stock prices we want
the percentile difference
01:01:02.190 --> 01:01:03.535
to be normally distributed.
01:01:11.930 --> 01:01:15.150
In other words, you want to
find the distribution of S_t
01:01:15.150 --> 01:01:24.230
such that the difference
of S_t divided by S_t
01:01:24.230 --> 01:01:25.500
is a normal distribution.
01:01:25.500 --> 01:01:27.250
So it's like a Brownian motion.
01:01:29.770 --> 01:01:31.607
That's the differential
equation for it.
01:01:41.560 --> 01:01:45.520
So the percentile difference
follows Brownian motion.
01:01:45.520 --> 01:01:46.572
That's what it's saying.
01:01:49.740 --> 01:01:54.259
Question, is S_t
equal to e sub B_t?
01:01:59.580 --> 01:02:05.020
Because in classical calculus
this is not a very absurd thing
01:02:05.020 --> 01:02:05.840
to say.
01:02:05.840 --> 01:02:08.800
If you differentiate each
side, what you get is dS_t
01:02:08.800 --> 01:02:13.110
equals e to the B_t, times dB_t.
01:02:13.110 --> 01:02:15.140
That's S_t times dB_t.
01:02:18.020 --> 01:02:20.080
It doesn't look that wrong.
01:02:20.080 --> 01:02:24.710
Actually, it looks
right, but it's wrong.
01:02:24.710 --> 01:02:28.710
For reasons that you
don't know yet, OK?
01:02:28.710 --> 01:02:33.740
So this is wrong
and you'll see why.
01:02:33.740 --> 01:02:36.650
First of all, Brownian
motion is not differentiable.
01:02:36.650 --> 01:02:38.445
So what does it even
mean to say that?
01:02:42.630 --> 01:02:46.740
And then that means if you
want to solve this equation,
01:02:46.740 --> 01:02:49.950
or in other words, if you
want to model this thing,
01:02:49.950 --> 01:02:51.265
you need something else.
01:03:01.840 --> 01:03:04.878
And that's where Ito's
calculus comes in.
01:03:22.970 --> 01:03:24.972
OK.
01:03:24.972 --> 01:03:26.651
I'll try not to rush too much.
01:03:29.420 --> 01:03:48.700
So suppose-- now we're
talking about Ito's calculus--
01:03:48.700 --> 01:03:49.810
you want to compute.
01:04:08.990 --> 01:04:10.380
So here is a motivation.
01:04:10.380 --> 01:04:12.240
You have a function f.
01:04:12.240 --> 01:04:15.340
I will call it a very
smooth function f.
01:04:15.340 --> 01:04:16.800
Just think about
the best function
01:04:16.800 --> 01:04:19.420
you can imagine, like
an exponential function.
01:04:19.420 --> 01:04:23.230
Then you have a Brownian
motion, and then you
01:04:23.230 --> 01:04:24.220
apply this function.
01:04:24.220 --> 01:04:25.886
As an input, you put
the Brownian motion
01:04:25.886 --> 01:04:27.240
inside the input.
01:04:27.240 --> 01:04:29.070
And you want to
estimate the outcome.
01:04:32.990 --> 01:04:34.795
More precisely, you
want to estimate
01:04:34.795 --> 01:04:35.878
infinitesimal differences.
01:04:43.360 --> 01:04:45.260
Why will we want to do that?
01:04:45.260 --> 01:04:49.521
For example, f can be
the price of an option.
01:04:49.521 --> 01:04:53.090
More precisely, let
f be this thing.
01:04:57.770 --> 01:04:58.270
OK.
01:04:58.270 --> 01:04:59.020
You have some s_0.
01:05:02.190 --> 01:05:04.910
Up to s_0, the value
of f is equal to 0.
01:05:04.910 --> 01:05:10.520
After s_0, it's just
a line with slope 1.
01:05:10.520 --> 01:05:14.590
Then f of Brownian
motion is just
01:05:14.590 --> 01:05:19.510
the price exercise-- what
is it-- value of the option
01:05:19.510 --> 01:05:21.095
at the expiration.
01:05:23.600 --> 01:05:25.660
T is the expiration time.
01:05:28.750 --> 01:05:29.680
It's a call option.
01:05:33.592 --> 01:05:35.059
That's the call option.
01:05:42.400 --> 01:05:46.830
So if your stock at time T goes
over s_0, you make that much.
01:05:46.830 --> 01:05:50.210
If it's below s_0,
you'll lose that much.
01:05:50.210 --> 01:05:52.635
More precisely, you have
to put it below like that.
01:05:52.635 --> 01:05:56.864
Let's just do it like that.
01:05:56.864 --> 01:05:58.340
And it looks like that.
01:06:02.770 --> 01:06:04.860
So that's like a
financial derivative.
01:06:04.860 --> 01:06:06.700
You have an underlying
stock and then
01:06:06.700 --> 01:06:08.780
some function applies to it.
01:06:08.780 --> 01:06:12.130
And then what you have, the
financial asset you have,
01:06:12.130 --> 01:06:14.135
actually can be described
as this function.
01:06:14.135 --> 01:06:17.190
A function of an
underlying stock, that's
01:06:17.190 --> 01:06:21.030
called financial derivatives.
01:06:21.030 --> 01:06:22.550
And then in the
mathematical world,
01:06:22.550 --> 01:06:27.574
it's just a function applied to
the underlying financial asset.
01:06:27.574 --> 01:06:29.240
And then, of course,
what you want to do
01:06:29.240 --> 01:06:31.280
is understand the
difference of the value,
01:06:31.280 --> 01:06:36.500
in terms of the difference
of the underlying asset.
01:06:36.500 --> 01:06:40.700
If B_t was a very
nice function as well.
01:06:40.700 --> 01:06:50.790
If B_t was differentiable, then
the classical world calculus
01:06:50.790 --> 01:06:59.085
tells us that d of f is equal to
d of B_t over d of t times dt.
01:07:04.530 --> 01:07:05.130
Yes.
01:07:05.130 --> 01:07:08.740
So if you can differentiate
it over the time difference,
01:07:08.740 --> 01:07:09.860
over a small time scale.
01:07:09.860 --> 01:07:13.920
All we have to do is
understand the differentiation.
01:07:13.920 --> 01:07:16.530
Unfortunately, we can't do that.
01:07:16.530 --> 01:07:21.530
We cannot do this.
01:07:21.530 --> 01:07:23.990
Because we don't know
what-- we don't even
01:07:23.990 --> 01:07:25.134
have this differentiation.
01:07:30.051 --> 01:07:30.550
OK.
01:07:34.880 --> 01:07:39.590
Try one, take one
failed, take two.
01:07:39.590 --> 01:07:42.220
Second try, OK?
01:07:42.220 --> 01:07:44.230
This is not
differentiable, but still I
01:07:44.230 --> 01:07:46.830
understand the minuscule
difference of dB_t.
01:07:46.830 --> 01:07:51.920
So what about this?
01:07:51.920 --> 01:07:55.210
df-- maybe I didn't
write something,
01:07:55.210 --> 01:08:08.550
f prime-- is equal to
just dB_t of f prime.
01:08:17.708 --> 01:08:19.180
OK?
01:08:19.180 --> 01:08:21.370
What is this?
01:08:21.370 --> 01:08:24.899
We can't differentiate
Brownian motion,
01:08:24.899 --> 01:08:28.040
but still we understand the
minuscule and infinitesimal
01:08:28.040 --> 01:08:30.630
difference of the
Brownian motion.
01:08:30.630 --> 01:08:34.140
So I just gave up trying to
compute the differentiation.
01:08:34.140 --> 01:08:38.060
But instead, I'm going to just
compute how much the Brownian
01:08:38.060 --> 01:08:44.040
motion changed over this small
time scale, this difference,
01:08:44.040 --> 01:08:47.029
and describe the
change of our function
01:08:47.029 --> 01:08:49.479
in terms of the differentiation
of our function f.
01:08:49.479 --> 01:08:53.119
f is a very good function,
so it's differentiable.
01:08:53.119 --> 01:08:53.785
So we know this.
01:08:53.785 --> 01:08:55.990
This is computable.
01:08:55.990 --> 01:08:58.670
This is computable.
01:08:58.670 --> 01:09:02.359
It's the difference of Brownian
motion over a very small time
01:09:02.359 --> 01:09:04.580
scale.
01:09:04.580 --> 01:09:07.029
So that at least
now is reasonable.
01:09:07.029 --> 01:09:08.040
We can expect it.
01:09:08.040 --> 01:09:10.399
It might be true.
01:09:10.399 --> 01:09:12.104
Here, it didn't
make sense at all.
01:09:12.104 --> 01:09:16.470
Here, it at least make
sense, but it's wrong.
01:09:16.470 --> 01:09:19.284
And why is it wrong?
01:09:19.284 --> 01:09:21.529
It's precisely because of this.
01:09:24.590 --> 01:09:33.260
The reason it's wrong,
the reason it is not valid
01:09:33.260 --> 01:09:46.190
is because of the fact
dB squared equals dt.
01:09:46.190 --> 01:09:51.930
And let's see how this comes
into play, this factor.
01:09:51.930 --> 01:09:55.390
I think that will be the last
thing that we'll cover today.
01:10:03.190 --> 01:10:03.840
OK.
01:10:03.840 --> 01:10:06.500
So if you remember where
you got this formula from,
01:10:06.500 --> 01:10:07.840
you probably won't remember.
01:10:07.840 --> 01:10:12.640
But from calculus, this follows
from Taylor's expansion.
01:10:12.640 --> 01:10:21.310
f of t plus x, I'll say,
is equal to f of t plus
01:10:21.310 --> 01:10:28.000
f prime of t times x, plus
f double prime of t over 2,
01:10:28.000 --> 01:10:42.000
times x squared plus-- over 3
factorial x cubed plus-- df is
01:10:42.000 --> 01:10:43.411
just this difference.
01:10:48.710 --> 01:10:50.690
Over a very small
time increase, we
01:10:50.690 --> 01:10:53.395
want to understand the
difference of the function.
01:10:53.395 --> 01:10:55.698
That's equal to f
prime t times x.
01:11:02.000 --> 01:11:03.530
OK.
01:11:03.530 --> 01:11:08.950
In classical calculus we were
able to ignore all these terms.
01:11:12.630 --> 01:11:24.924
So in the classical world f(t+x)
minus f(t) was about f prime t
01:11:24.924 --> 01:11:27.890
times x.
01:11:27.890 --> 01:11:31.730
And that's precisely
this formula.
01:11:31.730 --> 01:11:33.765
But if you use Brownian
motion here-- so what
01:11:33.765 --> 01:11:38.000
I'm trying to say is if
B at some time t plus x,
01:11:38.000 --> 01:11:42.350
minus Brownian
motion B at time t,
01:11:42.350 --> 01:11:44.360
then let's just write
down the Taylor formula.
01:11:44.360 --> 01:11:48.840
We get f prime at B_t.
01:11:48.840 --> 01:11:55.010
x will be this difference,
B at t plus x minus B at t.
01:11:55.010 --> 01:11:58.980
That's like the
difference in B_t.
01:11:58.980 --> 01:12:02.200
So up to this much
we see this formula.
01:12:02.200 --> 01:12:08.010
And the next term, we
get the second derivative
01:12:08.010 --> 01:12:12.850
of this function over
2 and x squared, x
01:12:12.850 --> 01:12:15.070
plus this difference.
01:12:15.070 --> 01:12:19.870
So what we get is dB_t squared.
01:12:19.870 --> 01:12:22.570
OK?
01:12:22.570 --> 01:12:26.370
But as you saw, this
is no longer ignorable.
01:12:26.370 --> 01:12:34.110
That is like a
dt, as we deduced.
01:12:34.110 --> 01:12:37.210
And that comes into play.
01:12:37.210 --> 01:12:46.090
So the correct-- then by
Taylor expansion, the right way
01:12:46.090 --> 01:12:54.020
to do it is df is equal to the
first derivative term, dB_t,
01:12:54.020 --> 01:13:01.168
plus the second derivative
term, double prime over 2 dt.
01:13:06.086 --> 01:13:07.210
This is called Ito's lemma.
01:13:15.812 --> 01:13:17.520
And now let's say if
you want to remember
01:13:17.520 --> 01:13:20.251
one thing from the math part,
try to make it this one.
01:13:25.130 --> 01:13:26.080
This had great impact.
01:13:31.330 --> 01:13:34.010
If you follow the
logic it makes sense.
01:13:39.777 --> 01:13:44.850
It's really amazing how somebody
came up with for the first time
01:13:44.850 --> 01:13:48.250
because it all makes sense.
01:13:48.250 --> 01:13:52.950
It all fits together if you
think about it for a long time.
01:13:52.950 --> 01:13:58.440
But actually, I once
saw that Ito's lemma
01:13:58.440 --> 01:14:01.790
is one of the most cited
lemmas, like most cited paper.
01:14:01.790 --> 01:14:05.030
The paper that's
containing this thing.
01:14:05.030 --> 01:14:06.575
Because people think
it's nontrivial.
01:14:09.420 --> 01:14:11.110
Of course, there
are facts that are
01:14:11.110 --> 01:14:13.160
being used more than
this, classical facts,
01:14:13.160 --> 01:14:15.590
like trigonometric functions,
exponential functions.
01:14:15.590 --> 01:14:17.732
They are being used
a lot more than this,
01:14:17.732 --> 01:14:19.940
but people think that's
trivial so they don't cite it
01:14:19.940 --> 01:14:22.430
in their research and paper.
01:14:22.430 --> 01:14:25.280
But this, people
respect the result.
01:14:25.280 --> 01:14:27.390
It's a highly nontrivial result.
01:14:27.390 --> 01:14:31.330
And it's really amazing how
just by adding this term,
01:14:31.330 --> 01:14:37.170
all this theory of calculus
all now fit together.
01:14:37.170 --> 01:14:41.300
Without this-- maybe it's
a too strong statement--
01:14:41.300 --> 01:14:47.270
but really Brownian motion
becomes much more rich
01:14:47.270 --> 01:14:48.400
because of this fact.
01:14:48.400 --> 01:14:49.796
Now we can do calculus with it.
01:14:53.540 --> 01:14:56.205
So there's two
things to remember.
01:14:56.205 --> 01:14:58.705
Well, if you want to remember
one thing, that's Ito's lemma.
01:14:58.705 --> 01:15:00.163
If you want to
remember two things,
01:15:00.163 --> 01:15:04.803
it's just quadratic variation,
dB_t squared is equal to dt.
01:15:08.590 --> 01:15:12.130
And I remember that's
exactly because B_t
01:15:12.130 --> 01:15:16.150
is like a normal
variable with 0, t.
01:15:16.150 --> 01:15:19.343
And time scale-- B_t is like
a normal random variable 0, t.
01:15:19.343 --> 01:15:22.120
dB_t squared is like
the variance of it.
01:15:22.120 --> 01:15:25.760
So it's t, and if you
differentiate it, you get dt.
01:15:25.760 --> 01:15:27.605
That was exactly
how we computed it.
01:15:30.400 --> 01:15:33.240
So, yeah, I'll just quickly
go over it again next time
01:15:33.240 --> 01:15:36.300
just to try to make it
stick in to your head.
01:15:36.300 --> 01:15:38.430
But please, think about it.
01:15:38.430 --> 01:15:40.980
This is really cool stuff.
01:15:40.980 --> 01:15:43.080
Of course, because
of that computation,
01:15:43.080 --> 01:15:46.290
calculus using Brownian motion
becomes a lot more complicated.
01:15:50.310 --> 01:15:53.370
Anyway, so I'll see
you on Thursday.
01:15:53.370 --> 01:15:56.770
Any last minute questions?
01:15:56.770 --> 01:15:57.947
Great.