WEBVTT
Kind: captions
Language: en
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This is a stochastic processes model 7, Brownian
motion and its applications. Lecture 2; processes
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derived from Brownian motion.
In the lecture one we have discussed the definition
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and properties of Brownian motions we started
with the random walk. Then the sample path
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of a random walk. Then we have given few properties
of random walk followed by that we have made
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the derivation of Brownian motion through
the random walk. Then we have discussed the
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sample path of a Brownian motion. Followed
by that we have discussed a few important
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properties such as strict sense stationary
increment, independent increment property,
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nowwhere differentiable property, self-similar
property, Markov property, between Gaussian
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process and the Brownian motion. Also we have
discussed the Kolmogorov equation and we have
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given the connection with the heat equation.
Then we have discussed the joint distribution
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of Poisson process also and finally we have
discussed the Martingale property of Brownian
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motion. So in that lecture 1 of module 7 was
completed.
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In the lecture 2 we are going to cover the
definition of a geometric Brownian motion.
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Then few properties also going to be discussed
followed by that we are going to discuss the
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applications of geometric Brownian motion
and also few examples of geometric Brownian
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motion.
Finally the process derived from Brownian
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motion that's a levy process also going to
be discussed.
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The definition of geometric Brownian motion.
A stochastic process Xt is said to be a stochastic
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process X of t where t is varies from 0 to
infinity is said to be geometric Brownian
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motion if X of t is of the form X naught exponential
of W where Wt is a Brownian motion. That means
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that whenever you have Brownian motion then
you make another stochastic process as a function
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of Brownian motion but it's of the form X
of t is equal to X of 0 e power Wt then the
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X of t is stochastic process is said to be
geometric Brownian motion. You know the range
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of Wt that is minus 3 infinity to infinity
since X of t is of the form X of 0 e power
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Wt therefore the range of Xt will be 0 to
infinity. The range of Xt is 0 to infinity.
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Therefore you can use this as a model for
the stock price at any time t like that you
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can go for modeling any pricing of any security
or derivatives at time t. So the Xt can be
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directly used in the application of finance.
You can see the sample path over the time
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the Xt here the range is from 0 to infinity
so you can see the continuous functions. You
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know that the sample path of Wt is a continuous
function and the Xt that is a geometric Brownian
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motion this is also a continuous function.
[Indiscernible] [00:05:01] W of t.
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Now we are going to discuss the very important
property that is the Markov property of the
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geometric Brownian motion. The geometric Brownian
motion is of the form X of t is equal to X
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naught e power Wt therefore for any H greater
than 0 you can write X of t plus HbA1c is
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same as X naught e power in W of t plus h.
I'm going to verify the Markov property therefore
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what I am going to do I'm going to add the
Wt and subtract Wt in the exponential. Therefore
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the next step will be X naught e power Wt
+ W of t minus h minus Wt. Already we we know
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that X of t is equal to X naught E power Wt
therefore X naught e power Wt can be replaced
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by X of t. Hence, X of t plus 1 is same as
X of t multiplied by e power Wt + h. That
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means the stochastic process at the time point
t plus h is same as at the time point t multiplied
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by exponential of the increment between the
time points t to t plus h in W. We already
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know that the Brownian motion or Wiener process
satisfies the Markov property and also the
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increments are independent along with increments
are stationary here we are going to use independent
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increments are independent. That means since
the Brownian motion has independent increments
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given Xt the future X of t plus h only depends
on the future implements of Brownian motion
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given Xt the X of t plus 1 t plus h it depends
only on W of t plus h minus W of t but since
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W is the Brownian motion, the Brownian motion
increments are independent therefore W of
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t plus h minus Wt is independent of 0 to W.
So this feature is independent of the the
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positive intergers. Here the positive integers
is from 0 to small time. Hence, the Markov
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property satisfies because the future depends
only on the present not the past. Therefore
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the Markov property is satisfied hence then
W the Xt is the [Indiscernible] [00:08:16]
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So this is the above result is valid for all
h greater than 0 therefore the Markov property
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is satisfied instance the future depends only
on the present not on the past or it is independent
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of the past. Hence the Markov property is
satisfied by Xt hence the Xt is a Markov process.
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So the Brownian motion is also a Markov process
geometric Brownian motion is a Markov process
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also geometric Brownian motion is also a [Indiscernible]
[00:08:56].
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we are going to find out what is the standard
[Indiscernible] [00:09:05] Brownian. We can
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use the moment generating function also. Here
the moment generating function of a normal
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distribution random variable X with the mean
mu the variance is Sigma square is given by
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for any random variables which is normally
distributed with the mean mu and variance
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[Indiscernible] [00:09:30] the moment generating
function it is M x of s as a function of s
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that is nothing but expectation of e power
s times x. So this expectation is exist because
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the random variable X is normally distributed
so the moment generating function exist because
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of the moments of all order is exists. So
that is same as a you can do it separately
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this calculation. So here I'm using the result
of moment generating function of normal distribution
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that is same as exponential of because it's
a function of [Indiscernible] [00:10:13] that's
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why it is mu times s plus half Sigma square
s square where s can take the value from minus
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infinity to infinity.
Why we are using the moment generating function
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because here the geometric Brownian motion
and the Brownian motion is the connected in
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the form of X of t is X of 0 e power Wt where
Wt is a normally distributed with the mean
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0 and the variance t. Hence I'm using the
moment generating function of the X of t as
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a function of s that is expectation of e power
X times Xt. That is same as since Wt is normally
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distributed with the mu times t Sigma square
t for a standard normal distribution the mu
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is 0 and the variance is 1. So here you can
make out that is same as exponential of the
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mu ts plus half Sigma square t s square by
replacing - by using the same the above logic
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here. [Indiscernible] [00:11:38] of this problem.
So sorry, there is a mistake. It is moment
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generating function of Wt and e of X times
Wt because X is normally distributed here
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it should not be exchanged with Wt. Here also
Wt therefore moment generating function of
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Wt is same as expectation of e power s times
Wt [Indiscernible] [00:12:09] now using s
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equal to 1 and 2 now we are finding the mean
and the variance of geometric Brownian motion
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because the mean of Xt is X naught times the
moment generating function of X of t at 1.
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So we are using the moment generating function
therefore we can get the mean of the geometric
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Brownian motion. Similarly finding the second
order moment we can get the variance of geometric
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Brownian motion.
After simplification that means first we have
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to use what is the moment generating function
of Wt and we have to use the form Xt is equal
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to X naught e power Wt using that you can
get that and after doing some simplification
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we get mean and variance of geometric Brownian
motion that is a expectation of Xt is at X
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naught times e power mu plus Sigma square
by two times t and first we can get the expectation
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of second order moment [Indiscernible] [00:13:22]
and the variance is equal to expectation of
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[Indiscernible] [00:13:24] minus expectation
of [Indiscernible] [00:13:27] then you can
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get the variance of Xt. So the variance of
Xt is of the form X naught whole square e
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power 2 mu plus Sigma square t multiplied
by e power Sigma square t minus 1. By substituting
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mu plus Sigma square by 2 as a power cap upon
bar you will get expectation of X of t is
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equal to X naught times e power [Indiscernible]
[00:14:04] t. More generally you can go for
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expectation of Xt divided X of s that is same
as e power power bar t - s. So in this way
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we are finding mean and variance of geometric
Brownian motion using the moment generating
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motion of normal distributed randomly.