We are moving into the next example, evaluate the Ito integral, ?_0^T¦?W(t)dW(t)? that
means integrant is W(t), integration with respect to W(t).
So you can use the partitionin the interval 0 to Tinto n pieces,n parts.Then as limit
ntends to infinity, this integration is nothing but lim-(n?8)?
?_(i=0)^(n-1)¦?W(ti)? and the difference ofWs.
So the same definition we have given in the Ito integrals also.
So here, the Ito integral is well defined because W(t) that is integral which is adapted
and alsothe mean square integral.
Therefore, this is Ito integral.
So the I(t) = ?_0^T¦?W(t)dW(t)? that integral is Ito integral.Buthere, we are going for
the upper limit is T, not the variable limit.
So for each I, the way we have writtenlim-(n?8) summation with the of w Ws, I and Ws, and
this is a W at the point ti whereas this one is a W(ti+1) - W(ti)).
Therefore, these two are thenon-overlapping Ws.So the increments areindependent.Therefore,W(ti)
and (W(ti+1) - W(ti)) are the independentrandom variables or having the normaldistributions.
We are using the propertyof Brownian motion therefore, W(ti) and (W(ti+1) - W(ti)) are
independent random variables.
So what Iam going to do here, letp be the set ofall finite subdivisions of pof theinterval
0 to T.Therefore,first we'll find out what is the Qpthat isdifferent from what or what
we wantis different.What wewant is lim-(n?8)?
?_(i=0)^(n-1)¦?W(ti)? with the difference.
But what we are defining now, Qpis nothing butthe difference of whole square.
but whatwe want is this W(ti) with W(ti+1) - W(ti).
So we started with Qpthat is equal to the differenceof whole square.The difference of
wholesquare is the same as suppose youtreat this as a, this as b.
So (a-b)2is same as a2-b2-2(a-b)-2b(a-b).So if you simplify, you'll get (a-b)2.
But if you expand this summationnow, this will be the only the last termwill be there
and all other termsvanishes.So therefore, you will (W(T))2-(W(0))2.
The last term and the first term willexist and all other terms will vanishbecause of
minus squares.
Whereas this one, -2?_(i=0)^(n-1)¦?W(ti)?(W(ti+1)-W(ti)).
That's what we want as n-->8.
But we knowthe result, limitnorm of p-->0 ofQpwill be T. That we have discussed in the
lecture 1, whenwediscussed the quadratic variation ofBrownian motion.In the lecture 1, wehave
discussed quadratic variation ofBrownian motion in that Qpisdefined limit norm of p-->0 Qpwill
be T. Therefore, what we want is a limit n-->8 of
this quantity and that is inthe third term in this Qp.
So now youapply a limit n-->8 in equation; so the left-hand sidebecomes the T, the right-handbecomes
(W(T))2- (W(0))2you know that W(0) is 0 for a standard Brownian motion.
Therefore, we will get only thefirst term; second term will be 0, thirdterm is the unknown.That's
the integration.
n-->8 of summation that is nothing but the I(T).
Therefore, ?_0^T¦?W(t)d(t)? that is nothing but in the left side we got T, therefore,
we will get (W(T))2-T is equal to, since it is -2 times, therefore it will be right.
Hence, the integration ?_0^T¦?W(t)d(t)?
becomes ((W(T) )^2-T)/2.
So here we have used the quadratic variation of Brownian motion is T that we have used
for the the interval 0 to T. Therefore, the integration 0 to T so we have
used the Brownian motion quadratic variation between the interval 0 to T that is T. Therefore,
the integration is going to be ((W(T) )^2-T)/2.
Hence, this integration, ((W(T) )^2-T)/2, we see that unlike Reimann integral, we have
extra term T. If the integration is with respect to boundedvariation differentiable function
thenyou won't have the term T here.
Just it is ((W(T) )^2)/2.
Suppose you replace W(s) by x, therefore you'll get
?_0^T¦?X(x)d(x)? that is nothing but x2/2 whenever the integrant is boundedvariation
as well as differentiablefunction.That is nothing but the Riemannintegral.So in the
Riemann integral youwon't have the extra term T/2, but here we have the extra term minus
T/2.
So this arises from the quadraticvariation of a Brownian motion which isfinite whereas
if it is a real valuefunction which is bounded then thequadratic variation is going to be
0.Buthere, the quadratic variation which isfinite value, therefore you are gettingminus -T/2.
So the nextremark, the above integral is defined notonly for the upper limit integration, T,
but also every upper limit ofintegration between 0 to T.Inthis integration, we have done the
upperlimit as T, therefore we used the quadratic variation of Brownian motion between the interval
0 to T thatis T. But instead of that, we cango for variable,t, also.
In that case, here you have to use the quadraticvariation of Brownian motion between theinterval 0 to
t and that will be t itself.
That's a variable t.
Therefore, thisintegration will be ?_0^t¦?W(s)dW(s)? is nothing but s is nothing but ((W(T) )^2-T)/2
where both the ts are small variable.
You see the construction of Ito integralis similar to one of the Stieltjes integration,
but instead of integratingwith respect to the deterministicfunction theIto integral
with respect tothe random function most precisely pathof Weiner process.So that is adifference
between the Ito integral andthe usual integral.The usual integral iswith respect to the deterministicfunction
whereas in the Ito integral, theintegration with respect to the path ofBrownian motion
or Weiner process whichis unbounded variation and nowheredifferential.
Therefore, the whole integration is different.
Consider this stochastic integral.
In this, first integral is a Riemannintegral of stochastic integrant whereasthe second
integral is aIto integral.
This equation is very useful in finance.
In the Black Scholemodel for pricing, the stock price is seeming to followgeometricBrownian
motion.
Now we are discussing the few propertiesof Ito integrals.Still we haveused a few properties
in the examples, but now you can understand how theproperties will be used in the examples.
The following resultsholds by the Itointegral defined in the equation 1.
Inthe equation 1, we have discussed - yeah, this is the Ito integral which we havediscussed
in the first equation.So thiswe are referring.
Yes, the first property, the integral I(t) T is a martingale with respect to thefiltration
F(t).
This is a veryimportant property.If you have a Itointegral that means the integrant isadopted
and the mean square integraland the Ito integral is a stochasticprocess and that stochastic
process ishaving the martingale property withrespect to the filtration F(t).
The same F(t) the integrant is adoptedalso.With respect to the same filtration, the integrant
is adoptedalso.
So the proof is if youwant to verify the stochastic processes martingale you have to check threeproperties;
the first property is it hasto be the stochastic processfor fixed T, it has to be integrable.
Then third property the conditionalexpectation has to satisfy the equality property.So if
these threeproperties are satisfied by thestochastic process then we say thestochastic process
has the martingale property.So here I(t) is a stochasticprocess for t=0.
So first, we have to verify it isintegrable.So you can find out theexpectation of I(t) or
every T that is going to be a finite value.Then the second one, I(t)has to be -- for fixed
T, I(t) has to be a F(t) measurablethat is adopted.So for fixed alreadythe integrant
is adopted, the integrationwith respect to W(T) and W(T) is alsoadapted to F(t) one can
prove, the I(t) is also adapted to the filtration F(t).
The third condition, expectation of I(t) given F(s)where s less than t that is same as I(s).
You have to prove expectation of I(t) given F(s) wheres is less than t that is equal to
I(s) if that is proved for all t.
Sincethese three properties of martingale is satisfied, I(t) will be a martingale withrespect
to the filtration F(t).
Here,I am not discussing the proof, here I am not giving the proof, but I havediscussed
whatever all the propertieshas to be verified.
Second property, expectation of I(t) that is same as aexpectation of 0 to t, integration
the, definition that will be 0.
Since it is a martingale it will have a constant mean.
Therefore, expectation of I(t) issame as expectation of I(0) and youknow how to evaluate expectation
of I(0) that will be 0.
And the third property, expectation of a (I(t))2 that is same as expectation of ?_0^t¦?X^2(s)ds?
that is same as ?_0^t¦?E(X^2(s))ds?So the expectation and the integration interchange
the place.
This is called Ito isometry.
So thisis a very important property of secondorder expectation that isthe integrant and earlier
you haveexpectation of (I(t))2 thatis nothing but ?_0^t¦?X^2ds? this is nothing but the
Reimann integral.
For fixed s, the expectation of X2(s) is the function of s.So you are integrating ?_0^t¦?E(X^2
(s)ds?.So this is nothing but the Reimann integral.
Now based on theproperty number 2 and 3, you can conclude Var(I(t)) is nothing but ?_0^t¦?E(X^2
(s)ds?because the E(I(t))=0, therefore Var(I(t)) = E(I(t)2)-E(I(t))2 that is
same as ?_0^t¦?E(X^2 (s)ds?.
One canfind the quadratic variation of I(t) also because I(t) is a stochasticprocess.So
you can find out thequadratic variation between the interval 0 to t.
So [I,I] between the interval 0 to t, that means a second ordervariation between the
interval 0 to t for the function I(t).
I(t) is a Itointegral whereas the quadratic variationis nothing but integration ?_0^t¦?X^2
(s)ds?.So this is the Reimann integral.
So I have now given the proof ofthe fifth property also.We are juststating the results
of Ito integral andwe have used a few properties in theexamples.
The Ito integral is a random variablefor all T, for all W belonging to O. Therefor,e you
can write down, I(t)(w) that is nothing but limit n-->8, the value X(si), the difference
of Ws.
You know that X is a random variable, thedifference of Ws is also a randomvariable.Therefore,
for fixed n, thissummation is equal to 0 to n-1 will be nothing but a random variable.
Weare finding limit n-->8 of this summation.Therefore,actually theconvergence takes place only for thesubsequence.That
means I(t) is arandom variable and that random variableis nothing but the convergence of
theright-hand side, limit n-->8 of this summation and itsintegral depends on the sample pathbecause
you are finding the differenceof W(si+1)-W(si) therefore, this integral depends onthe sample
path.
Also, we need these following properties the dW(t)dW(t) will be dt that is nothing but
the quadratic variation of Brownian motionis nothing, but a finite value if you arefinding
the increment between theinterval 0 to t then the quadraticvariation is small t, whereas
if it is areal value function, which is adifferentiable, then dtdt, the quadraticvariation of the function
t will be 0.
Andthe mixed variation, cross-variation dW(t)dt that will be 0.
So if you find out the quadratic variation withthe W(t) that will be the t, whereas a
quadraticvariation with the function t will be 0and the cross-variation will be 0.Sothis
result will be used when you arefinding the I(t).
Here are theimportant referencesfor the Ito integrals.In the next class, we will consider
the Ito formulas.