WEBVTT
Kind: captions
Language: en
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As, we have discussed in the last class, we
discussed symmetric random walk and how we
00:00:21.860 --> 00:00:27.980
can scale it slightly up and down and so that
we can make it more zigzag and go towards
00:00:27.980 --> 00:00:29.460
what is called a Brownian motion.
00:00:29.460 --> 00:00:35.760
So, the Brownian motion is a continuous stochastic
process which exhibits the property of a symmetric
00:00:35.760 --> 00:00:37.760
random walk.
00:00:37.760 --> 00:00:41.760
So, that is the idea of a Brownian motion.
00:00:41.760 --> 00:00:59.120
So, for example, if you look at the motion
of a pollen grain in water, it would be something
00:00:59.120 --> 00:01:02.749
like this, more zigzagging than this than,
I can draw.
00:01:02.749 --> 00:01:13.810
So, this is something a motion of a particle
in a gas chamber a motion of a molecule.
00:01:13.810 --> 00:01:19.540
So, Brownian motion encapsulates lot of phenomenon.
00:01:19.540 --> 00:01:25.250
So, what did I say, the best way to remember
about Brownian motion is that, Brownian motion
00:01:25.250 --> 00:01:30.500
is a continuous analog of symmetric random
walk.
00:01:30.500 --> 00:01:36.140
In a continuous setting, it behaves in the
way a symmetric random walk behaves in the
00:01:36.140 --> 00:01:37.490
discrete setting.
00:01:37.490 --> 00:01:44.890
So as usual, I will have the probability space
which is written down here.
00:01:44.890 --> 00:02:12.470
So, Brownian motion, so, your stochastic process,
Wt some people write W lower index t.
00:02:12.470 --> 00:02:15.190
it is up to you.
00:02:15.190 --> 00:02:54.670
it does not matter, is called a Brownian motion,
if given any
00:02:54.670 --> 00:03:15.130
time points, t is time actually, spreading
time points say tm tn whatever.
00:03:15.130 --> 00:03:23.350
Given any time points like this the increments
are independent.
00:03:23.350 --> 00:03:29.560
The increments mean just like the way you
have done increments in the symmetric random
00:03:29.560 --> 00:03:35.099
walk, the increments Wt1 minus Wt0.
00:03:35.099 --> 00:03:45.910
Wt0 is actually 0, W0 is 0 just a minute,
I think, I should just change it a bit.
00:03:45.910 --> 00:03:55.200
A stochastic process with W0 is equal to 0.
00:03:55.200 --> 00:03:56.989
So, it does not matter.
00:03:56.989 --> 00:04:03.720
Whatever be our scenario omega, W0 is always
0.
00:04:03.720 --> 00:04:06.920
So, it is identically a zero function.
00:04:06.920 --> 00:04:08.290
So, W0 itself is a function.
00:04:08.290 --> 00:04:11.970
Please remember it is a random variable.
00:04:11.970 --> 00:04:18.181
So, whatever be the form of that random or
whatever be this random variable, whatever
00:04:18.181 --> 00:04:23.220
be the scenario omega, W of that would always
be 0.
00:04:23.220 --> 00:04:31.560
That is the meaning of the whole thing.
00:04:31.560 --> 00:04:45.310
So, this Wt2 minus Wt1 this random variable,
these increments in the random variables.
00:04:45.310 --> 00:04:49.850
Hence, how much change you are having how
basically, you are telling that how much zigzagging
00:04:49.850 --> 00:04:59.170
has taken place in the interval t2 to t1,
t0 to t1, t1 to t2, t2 to t3 and so forth.
00:04:59.170 --> 00:05:05.550
So, it is some sort of a broad measure of
the zigzagging that has taken place.
00:05:05.550 --> 00:05:08.600
How much the function value has changed at
the 2 ends.
00:05:08.600 --> 00:05:28.940
These are independent random variables.
00:05:28.940 --> 00:05:31.810
Now the independent random variables.
00:05:31.810 --> 00:05:36.020
This is one property you know from the property
of the symmetric random walk.
00:05:36.020 --> 00:05:40.601
Also, you know the symmetric random walk,
the expectation of increments are 0 and the
00:05:40.601 --> 00:05:46.270
variance of the increments are the difference
between the 2 time end points.
00:05:46.270 --> 00:05:53.240
Here, actually one can prove through the Central
Limit Theorem that, these increments actually
00:05:53.240 --> 00:05:57.540
follow normal distribution, which we do not
prove because, that would take too much of
00:05:57.540 --> 00:06:00.370
time, it is a very short and compact course.
00:06:00.370 --> 00:06:16.900
So, these are independent random variables
and Wti plus 1 minus Wti, this random variable
00:06:16.900 --> 00:06:28.199
follows normal distribution, with mean 0 and
variance ti plus 1 minus ti.
00:06:28.199 --> 00:06:35.759
So, if all these properties, 2 properties
are followed, first that this has to be maintained
00:06:35.759 --> 00:06:42.169
whatever be the scenario, whatever be your
time points increasing, if you take an increasing
00:06:42.169 --> 00:06:50.970
set of time points finite set of time points,
then these things are independent and these
00:06:50.970 --> 00:06:57.520
increments themselves follow normal distribution,
for every i.
00:06:57.520 --> 00:07:10.300
Of course, this has to be for every i, now
start from i equal to 0 to i equal to n, i
00:07:10.300 --> 00:07:15.220
equal to n minus 1 basically.
00:07:15.220 --> 00:07:22.270
So, each of these, is following a normal distribution.
00:07:22.270 --> 00:07:28.720
So, this is what is called Brownian motion.
00:07:28.720 --> 00:07:30.410
Does stock price follow Brownian motion?
00:07:30.410 --> 00:07:35.040
The stock price truly does not follow Brownian
motion, though it looks like one.
00:07:35.040 --> 00:07:40.090
Because Brownian motions can take negative
values, because the symmetric random walks
00:07:40.090 --> 00:07:44.419
can also take negative values, but a stock
price can never take negative value.
00:07:44.419 --> 00:07:50.830
Once a stock price is 0, that stock has to
get out of the market now it is a 0-value
00:07:50.830 --> 00:07:51.830
stock.
00:07:51.830 --> 00:07:58.440
So, stock prices are not really a model using
Brownian motion, but actually these stock
00:07:58.440 --> 00:08:04.280
prices actually were modeled by Bachelier
in 1900, who was the student of Henry Poincare
00:08:04.280 --> 00:08:11.789
using Brownian motion and showing that the
pricing of such commodities in the stock market,
00:08:11.789 --> 00:08:16.350
of various instruments in the stock market
can be achieved by solving the heat equation.
00:08:16.350 --> 00:08:26.890
So, he linked the processing the financial
markets to partial differential equations.
00:08:26.890 --> 00:08:37.589
And of course, his idea was lost and now later
on he has become a very famous name.
00:08:37.589 --> 00:08:48.510
But the interesting part is that, the way
stock market or movements of stock prices
00:08:48.510 --> 00:08:53.780
are modeled that is called a geometric Brownian
motion and then that is built using, that
00:08:53.780 --> 00:08:59.280
will always give you a nonnegative thing,
which is built using a Brownian motion and
00:08:59.280 --> 00:09:05.250
you can understand and taking the trick that,
if you want to get non-negativity always use
00:09:05.250 --> 00:09:07.320
the exponential function.
00:09:07.320 --> 00:09:14.940
So, that trick is being played in this area
pretty often.
00:09:14.940 --> 00:09:22.130
So, now we will introduce what is called,
if I want to define a filtration associated
00:09:22.130 --> 00:09:39.920
with a Brownian motion, how do I define a
filtration associated with a Brownian motion?
00:09:39.920 --> 00:10:01.450
So, what I will do in this filtration associated
with the Brownian motion.
00:10:01.450 --> 00:10:25.040
So, this filtration Ft is a collection of
Sigma-algebras, this is a collection such
00:10:25.040 --> 00:10:50.860
that, number one, whenever t is strictly bigger
than s F of s must always be contained in
00:10:50.860 --> 00:10:55.740
Ft, but you have information as time evolves.
00:10:55.740 --> 00:11:04.019
So, this should always be contained in Ft
okay.
00:11:04.019 --> 00:11:15.680
Now number two, is that Wt must be adapted
to the filtration.
00:11:15.680 --> 00:11:36.910
The Brownian motion must be adapted to the
filtration
00:11:36.910 --> 00:12:11.700
and the third point is, so if you have this
situation then, Wu minus Wt is independent
00:12:11.700 --> 00:12:17.649
of Ft.
00:12:17.649 --> 00:12:22.839
That is, once you move beyond t, Ft does not
have any information about this.
00:12:22.839 --> 00:12:29.600
This is independent of what information you
have in Ft.
00:12:29.600 --> 00:12:36.110
Ft can also be viewed in some sense as a smallest
Sigma-algebra generated by the stochastic
00:12:36.110 --> 00:12:38.279
process till a given time.
00:12:38.279 --> 00:12:47.620
So, take all the values of the random variables
omega t, up to a given time and take the Sigma-algebra
00:12:47.620 --> 00:12:54.130
generated by them, that can be also viewed
as one of the filtrations right.
00:12:54.130 --> 00:12:59.990
You could have larger filtrations also but
that is essentially it is.
00:12:59.990 --> 00:13:06.410
So, once we know that that we can have some
filtration defined like this, we can prove
00:13:06.410 --> 00:13:16.440
that the Brownian motion is a Martingale.
00:13:16.440 --> 00:13:55.220
So, again, just do like this, take t less
than equal to s, so you will find that the
00:13:55.220 --> 00:13:56.560
tricks are almost similar.
00:13:56.560 --> 00:14:18.250
So, what I do is, I, and now I breakup the
whole thing.
00:14:18.250 --> 00:14:35.390
You know, Ws is completely determined by Fs
is known.
00:14:35.390 --> 00:14:37.970
So, you have to take out what is known.
00:14:37.970 --> 00:14:43.950
So, Ws is equal to Ws into 1 you take out
the Ws expectation of 1 of constant random
00:14:43.950 --> 00:14:46.940
variable is the same so.
00:14:46.940 --> 00:15:00.700
Now here, Wd minus Ws as the third definition,
does not depend on Fs.
00:15:00.700 --> 00:15:07.400
So, it is nothing bu,t it is so there is no
conditional expectation here, so is just the
00:15:07.400 --> 00:15:14.839
expectation, so this random variable is just
this constant random variable plus Ws.
00:15:14.839 --> 00:15:22.430
You take the Ws out, expectation of 1 given
Fs expectation of 1 is 1 basically that is
00:15:22.430 --> 00:15:23.430
it.
00:15:23.430 --> 00:15:28.649
Because 1, just the constant random variable
does not depend on F s but this you know is
00:15:28.649 --> 00:15:41.430
0 because of this fact, 0 plus W s which is
equal to W s and that proves that it is a
00:15:41.430 --> 00:15:42.430
Martingale.
00:15:42.430 --> 00:15:50.769
There are several other Martingales, which
are associated with the Brownian motion.
00:15:50.769 --> 00:15:55.790
A Martingale that we are going to write now,
is very-very important in finance specifically
00:15:55.790 --> 00:15:59.010
in calculation of risk neutral probabilities
and all those things.
00:15:59.010 --> 00:16:11.050
So, this is a very important Martingale called
the exponential Martingale.
00:16:11.050 --> 00:16:28.290
So, the exponential Martingale
is defined like this.
00:16:28.290 --> 00:16:39.390
So, exportation means, e to the power, so
sigma Wt, where sigma is positive number,
00:16:39.390 --> 00:16:50.070
exponential means e to the power of this.
00:16:50.070 --> 00:16:55.170
Of course, this itself is a random variable
because you are taking exponential to the
00:16:55.170 --> 00:16:56.459
power of a random variable.
00:16:56.459 --> 00:17:04.500
So, it is, when you are taking the exponential,
exponentiated by a random variable.
00:17:04.500 --> 00:17:06.740
So, this itself is a random variable.
00:17:06.740 --> 00:17:15.270
Now the idea is that if you have a filtration
associated with respect to Wt then that same
00:17:15.270 --> 00:17:21.559
filtration will be associated with Zt, because
if Wt is adapted to a given filtration Zt
00:17:21.559 --> 00:17:27.210
will also be adapted to the same filtration,
because knowledge of Zt singularly depends
00:17:27.210 --> 00:17:30.609
on the knowledge of Wt right.
00:17:30.609 --> 00:17:33.871
The question is whether it is a Martingale.
00:17:33.871 --> 00:17:36.519
So, one has to prove that this is also a Martingale.
00:17:36.519 --> 00:17:41.360
So, we start in the similar fashion.
00:17:41.360 --> 00:17:45.239
This proof is not as straightforward, as this
proof, though similar type of approaches would
00:17:45.239 --> 00:18:01.070
be used but let us just go and do it.
00:18:01.070 --> 00:18:14.409
This omega, the sigma that you see, this we
will finally talk about as volatility of the
00:18:14.409 --> 00:18:15.409
stock price movement.
00:18:15.409 --> 00:18:20.889
It captures, how the randomness, captures
the massive movement.
00:18:20.889 --> 00:18:27.629
It really gives you a feel of the zigzagging
of the path how the prices are how fast they
00:18:27.629 --> 00:18:37.509
are going and coming down so that sort of
so this captures that idea basically.
00:18:37.509 --> 00:18:47.110
So, F is a filtration associated with the
Brownian motion Wt.
00:18:47.110 --> 00:19:20.700
Say I have added sigma Ws and subtracted sigma
Ws from here.
00:19:20.700 --> 00:19:28.359
Observe that sigma square t is a nonrandom
part.
00:19:28.359 --> 00:19:35.610
It is not a random variable right.
00:19:35.610 --> 00:19:42.960
So, here is the product x into y okay.
00:19:42.960 --> 00:19:49.610
So, the question is, in this sort of situations
right, there are some deeper questions.
00:19:49.610 --> 00:19:57.379
If you go back to your original, where we
had written down the laws of conditional expectation,
00:19:57.379 --> 00:20:03.860
the rules then, we expected this and this,
this product has to be integrable because
00:20:03.860 --> 00:20:07.940
if you have to maintain the definition of
conditional expectation.
00:20:07.940 --> 00:20:13.759
Of course, product has to be integrable means
you need to define, what is the meaning of
00:20:13.759 --> 00:20:20.979
integration of these two random variables
okay.
00:20:20.979 --> 00:20:23.799
Is such random variables are integrable?
00:20:23.799 --> 00:20:33.259
That is the question we are not going to answer
right now.
00:20:33.259 --> 00:20:39.330
We will later on show that these are actually
integral, let alone discuss the integrability
00:20:39.330 --> 00:20:44.229
of this, that is they are actually integrable
random variables, because they will come out
00:20:44.229 --> 00:20:46.539
to be solution of certain equations.
00:20:46.539 --> 00:20:52.529
So, and if you have a Brownian motion, how
can you integrate it.
00:20:52.529 --> 00:20:59.809
Can you integrate it just like any other random
variable, can you find the expectation of
00:20:59.809 --> 00:21:02.330
a Brownian motion?
00:21:02.330 --> 00:21:04.279
The answer is yes.
00:21:04.279 --> 00:21:09.169
I can find the expectation of this Brownian
motion because this is 0.
00:21:09.169 --> 00:21:14.389
Can you find the expectation of this Brownian
motion?
00:21:14.389 --> 00:21:17.789
The answer is expectation of Brownian this
Brownian motion is this, minus half sigma
00:21:17.789 --> 00:21:18.789
square t.
00:21:18.789 --> 00:21:27.019
Now it will be left to the reader to decide
whether.
00:21:27.019 --> 00:21:30.269
Of course, there are certain little technical
issues, I am not getting into, but it is clear
00:21:30.269 --> 00:21:31.269
that expect.
00:21:31.269 --> 00:21:34.710
What is meant by the meaning of ntegrability
of a random variable?
00:21:34.710 --> 00:21:36.460
Integrability means that the expectation is
finite.
00:21:36.460 --> 00:21:42.320
The expectation of this is finite, this is
finite.
00:21:42.320 --> 00:21:52.509
So, this exponential function will have the
integral will be integrable and this exponential
00:21:52.509 --> 00:21:53.889
function would be also integrable.
00:21:53.889 --> 00:21:55.970
So, at the end if you look at this part.
00:21:55.970 --> 00:22:02.470
So, this is nothing but this part, but this
part this is because, this has mean minus
00:22:02.470 --> 00:22:07.929
half sigma square t because, this mean is
0 and, this exponential is nothing but a constant
00:22:07.929 --> 00:22:13.650
thing, so mean of this is nothing but minus
half sigma square t so this is a thing with
00:22:13.650 --> 00:22:14.650
finite mean.
00:22:14.650 --> 00:22:22.710
Now if you take a you are taking exponentiation
so basically you are taking expectation of
00:22:22.710 --> 00:22:23.710
the exponential function.
00:22:23.710 --> 00:22:28.679
If the random variable itself has a finite
mean, then if you take the exponential function,
00:22:28.679 --> 00:22:32.539
that exponential function will also have a
finite mean it will have finite expectation.
00:22:32.539 --> 00:22:36.919
That is why it is meaningful to apply the
fact that I can take out what is known.
00:22:36.919 --> 00:22:46.529
You see this part is known to me when, at
time s, this whole part is known to me because
00:22:46.529 --> 00:22:51.809
this only depend this is just the evolution
of W the random variable up to the time s.
00:22:51.809 --> 00:22:54.429
So, I will take this part out.
00:22:54.429 --> 00:22:56.769
Taking out what is known I will have exponential.
00:22:56.769 --> 00:23:03.570
So, it is very important to get certain technicalities
clear, before you move, where you are actually
00:23:03.570 --> 00:23:04.879
applying the results correctly.
00:23:04.879 --> 00:23:10.820
That is an important thing, that one needs
to learn as one goes on doing more mathematics.
00:23:10.820 --> 00:23:19.580
Sometimes we can do some hand waving but not
always.
00:23:19.580 --> 00:23:32.841
Now once, I have this, remember that this
thing is independent of Fs, which means, I
00:23:32.841 --> 00:24:08.919
just need to calculate this.
00:24:08.919 --> 00:24:15.580
Now what is this, sorry, I would not have
the Fs here sorry it is independent.
00:24:15.580 --> 00:24:21.320
Now if those, who know some probability, they
will understand that, I know that Wt minus
00:24:21.320 --> 00:24:29.519
Ws is a normal distribution with mean 0 and
variance t minus s and this is nothing but
00:24:29.519 --> 00:24:33.549
the moment generating function of a normal
distribution.
00:24:33.549 --> 00:24:38.769
I have not spoken about moment generating
function in this discussion, but if you forget
00:24:38.769 --> 00:24:45.029
about this term moment generating function,
you can directly compute this expectation.
00:24:45.029 --> 00:24:50.769
I am not going to compute for you, this will
come as a homework and this will appear in
00:24:50.769 --> 00:24:54.190
your assignments.
00:24:54.190 --> 00:24:59.260
So, what I am just writing down the answer.
00:24:59.260 --> 00:25:14.049
Even this is simple integration, so I am just
not going to do that.
00:25:14.049 --> 00:25:28.139
This is equal to exponential of half sigma
square t minus s okay.
00:25:28.139 --> 00:25:32.200
So, this is what you will have that is the
answer.
00:25:32.200 --> 00:25:41.049
So, once you write that down then, the final
answer would be the following, that if you
00:25:41.049 --> 00:25:50.389
write this as exponential so you write this
as exponential sigma Ws minus half sigma square
00:25:50.389 --> 00:26:02.119
t and then you write this as exponential of
sigma square t minus s, so this is nothing
00:26:02.119 --> 00:26:19.950
but exponential sigma Ws minus half sigma
square s so this is nothing but Zs.
00:26:19.950 --> 00:26:28.409
There is another Martingale which is helpful
in finance is the following.
00:26:28.409 --> 00:26:42.669
It is Zt is W square minus t.
00:26:42.669 --> 00:26:51.330
So, this will also go as an exercise in your
homework assignments, to prove that this is
00:26:51.330 --> 00:26:53.609
a Martingale.
00:26:53.609 --> 00:26:58.419
Now we will talk about, so what is happening.
00:26:58.419 --> 00:27:07.590
We will talk about how to compute the joint
probability of a Brownian motion at certain
00:27:07.590 --> 00:27:09.460
given time points.
00:27:09.460 --> 00:27:18.489
You take Wt at any t, what is the distribution
of this random variable.
00:27:18.489 --> 00:27:35.879
Wt can be always written as, Wt minus W0 and
this has normal 0t.
00:27:35.879 --> 00:27:46.049
So, if you want to know whether, at a given
time t your Wt is lying between some points
00:27:46.049 --> 00:28:06.989
a and b, it is obvious you can just use the
basic idea that is.
00:28:06.989 --> 00:28:12.909
The question is, suppose I have now n time
points which are greater than 0, 0 I know
00:28:12.909 --> 00:28:27.950
where it is, so with n time points and we
ask you the question, how to find sorry, say
00:28:27.950 --> 00:28:39.429
Wt1, how do I do it.
00:28:39.429 --> 00:28:53.559
So, let me look at the case at the very first,
this is how you compute what are called the
00:28:53.559 --> 00:28:56.139
transition probability densities let us see.
00:28:56.139 --> 00:29:04.460
So, now suppose, you are given the information
that, under the given scenario Wt1, under
00:29:04.460 --> 00:29:08.049
the given scenario omega Wt1 omega was x okay.
00:29:08.049 --> 00:29:17.000
So, suppose I know that it is known to me
that, at t1 for the given scenario W, at time
00:29:17.000 --> 00:29:22.059
t1 it was x where say x1, x1 lying between
b1 and a1.
00:29:22.059 --> 00:29:36.020
So, what I now want to do is, I want to calculate
the probability at W t2 lies between a2 and
00:29:36.020 --> 00:29:44.049
b2 given that Wt1 is x1.
00:29:44.049 --> 00:29:45.279
That is what I want to do.
00:29:45.279 --> 00:29:54.509
But, if you see, if I fixed Wt1 as x1 then,
what do I have.
00:29:54.509 --> 00:30:07.570
I have Wt2 minus x1 as my random variable,
x1 is known to me that, at time t1 x1 is what
00:30:07.570 --> 00:30:08.570
happened.
00:30:08.570 --> 00:30:14.929
So, now what is the probability that Wt2 would
lie between the value of Wt2 for the given
00:30:14.929 --> 00:30:17.899
scenario omega will lie between a2 and b2.
00:30:17.899 --> 00:30:21.929
So, for the scenario omega t1 I know what
will happen.
00:30:21.929 --> 00:30:23.700
It was at x1.
00:30:23.700 --> 00:30:29.519
Now find so it is in the state x1 so what
is the probability that, the next state would
00:30:29.519 --> 00:30:30.519
lie between a2 and b2.
00:30:30.519 --> 00:30:34.700
It is like sort of a computing transition
probabilities.
00:30:34.700 --> 00:30:45.289
So, here if you look at this thing, the expectation
of this, this is also normal random variable,
00:30:45.289 --> 00:30:57.019
so this also follows normal random variable,
but the expectation of this, is what, expectation
00:30:57.019 --> 00:31:16.109
is, so it is a, see here, we had, so this
new random variable z, so Wt2 so, is equal
00:31:16.109 --> 00:31:17.989
to z plus x1.
00:31:17.989 --> 00:31:22.959
So, under this given information the expectation
of Wt2.
00:31:22.959 --> 00:31:35.929
So, Wt2 under this information follows normal
x1 with, because this is the time t1, so this
00:31:35.929 --> 00:31:42.080
z follows normal 0 at time t2 t1, so Wt2 is
nothing but x1 plus z.
00:31:42.080 --> 00:31:49.620
So, Wt2 follows this one, because this expectation
is 0.
00:31:49.620 --> 00:31:54.099
So, once you know this fact then what would
happen.
00:31:54.099 --> 00:32:05.600
This means now I can compute, sorry, I should
be having, so we will be applying the same
00:32:05.600 --> 00:32:14.719
idea of conditional probability, that probability
of A Intersection B is probability of A into
00:32:14.719 --> 00:32:27.539
probability of B sorry, so probability of
A given B is probability of A intersection
00:32:27.539 --> 00:32:31.110
B by probability of B.
00:32:31.110 --> 00:32:32.570
This is what we had learnt.
00:32:32.570 --> 00:32:40.209
So, you can always write probability of A
Intersection B is probability of A given B
00:32:40.209 --> 00:32:44.989
into probability of B. That is exactly what
we are going to do here.
00:32:44.989 --> 00:32:49.309
So first we are computing the conditional
expectation and then using this we will write
00:32:49.309 --> 00:32:51.359
the joint probability.
00:32:51.359 --> 00:33:08.899
So, this conditional expectation is, 1 by
the conditional probability is, 1 by x2 is
00:33:08.899 --> 00:33:20.620
the variable that I am using for the variable
x2, x1 is what is the mean.
00:33:20.620 --> 00:33:35.179
Now, if I want to write the joint probability
so this is my conditional density.
00:33:35.179 --> 00:33:46.629
This is my conditional density function and
thus I can write the conditional density function
00:33:46.629 --> 00:33:49.830
in a more simpler way.
00:33:49.830 --> 00:33:55.649
So, I can write the conditional density function
in a compact form but let us do a general
00:33:55.649 --> 00:33:59.149
form and then we will tell you how to write
the conditional density function.
00:33:59.149 --> 00:34:31.389
If you look at this, now I am writing this
one, is a joint probability okay, and this
00:34:31.389 --> 00:34:48.029
is equal to, so first I will write, I will
tell you, what is the meaning of this.
00:34:48.029 --> 00:35:03.210
So, these are the marginal probabilities,
a marginal densities.
00:35:03.210 --> 00:35:08.630
So, the marginal density, in general is given
like this.
00:35:08.630 --> 00:35:16.170
So, what is the density function associated
with the fact my current state is y and then
00:35:16.170 --> 00:35:21.740
in time t, I will move to the state x, at
time 0 say if my current stated time 0 is
00:35:21.740 --> 00:35:28.990
y in time t, I will move to the state x, what
is the density function associated with that
00:35:28.990 --> 00:35:29.990
particular transition.
00:35:29.990 --> 00:35:38.930
So, this is called the transitional density
function or the conditional density, transition
00:35:38.930 --> 00:35:53.460
density and that is given as, one by root
2 pi t, e to the power of minus x minus y
00:35:53.460 --> 00:35:59.500
whole square by 2t .
00:35:59.500 --> 00:36:04.339
So, this is my conditional density function.
00:36:04.339 --> 00:36:10.829
So, I would keep it as an exercise for you
to write down for the case till end.
00:36:10.829 --> 00:36:12.380
How can you write it?
00:36:12.380 --> 00:36:19.160
So, here you see, from 0 my state was 0, it
is always 0, W0, 0.
00:36:19.160 --> 00:36:23.869
I have come to the state X1 at t1.
00:36:23.869 --> 00:36:32.170
Given that, I am in state x1 at t1, within
the time t1 to t2, I have, within the time
00:36:32.170 --> 00:36:35.269
span t2 minus t1, I have come to the state
x2.
00:36:35.269 --> 00:36:37.280
So, essential a marco process.
00:36:37.280 --> 00:36:41.000
Brownian motion is also a marco process those
who know about marco process.
00:36:41.000 --> 00:36:48.010
So, we are just trying to compute, this is
nothing but a conditional, this is a transitional
00:36:48.010 --> 00:36:49.010
probability actually.
00:36:49.010 --> 00:36:54.470
It tells you, given the state is at x1, what
is the probability that the state is between
00:36:54.470 --> 00:36:57.900
a2 and b2 so next state.
00:36:57.900 --> 00:37:02.299
Because it is a continuous thing, you cannot
say that probability Wt2 is x2 because that
00:37:02.299 --> 00:37:06.900
will become 0, so you just have to say what
is in between.
00:37:06.900 --> 00:37:09.080
So, that is the whole thing.
00:37:09.080 --> 00:37:14.760
So, I know what is at 0 I will come to the
state x1.
00:37:14.760 --> 00:37:21.819
So, the conditional density at x1 is the conditional
density at x2.
00:37:21.819 --> 00:37:27.839
I am at time t1 and I am at x1 and at time
t2, so within the time spend t2 minus t1 I
00:37:27.839 --> 00:37:31.420
have come to x2 then what is the conditional
density.
00:37:31.420 --> 00:37:36.910
So, the joint integration of this, would give
me the conditional probability, which can
00:37:36.910 --> 00:37:41.560
be motivated, is motivated from this very
basic idea.
00:37:41.560 --> 00:37:47.910
Thank you very much.