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