WEBVTT
Kind: captions
Language: en
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We are moving into the next example, evaluate
the Ito integral, ?_0^T¦?W(t)dW(t)? that
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means integrant is W(t), integration with
respect to W(t).
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So you can use the partitionin the interval
0 to Tinto n pieces,n parts.Then as limit
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ntends to infinity, this integration is nothing
but lim-(n?8)?
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?_(i=0)^(n-1)¦?W(ti)? and the difference
ofWs.
00:00:41.600 --> 00:00:47.629
So the same definition we have given in the
Ito integrals also.
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So here, the Ito integral is well defined
because W(t) that is integral which is adapted
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and alsothe mean square integral.
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Therefore, this is Ito integral.
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So the I(t) = ?_0^T¦?W(t)dW(t)? that integral
is Ito integral.Buthere, we are going for
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the upper limit is T, not the variable limit.
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So for each I, the way we have writtenlim-(n?8)
summation with the of w Ws, I and Ws, and
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this is a W at the point ti whereas this one
is a W(ti+1) - W(ti)).
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Therefore, these two are thenon-overlapping
Ws.So the increments areindependent.Therefore,W(ti)
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and (W(ti+1) - W(ti)) are the independentrandom
variables or having the normaldistributions.
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We are using the propertyof Brownian motion
therefore, W(ti) and (W(ti+1) - W(ti)) are
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independent random variables.
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So what Iam going to do here, letp be the
set ofall finite subdivisions of pof theinterval
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0 to T.Therefore,first we'll find out what
is the Qpthat isdifferent from what or what
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we wantis different.What wewant is lim-(n?8)?
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?_(i=0)^(n-1)¦?W(ti)? with the difference.
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But what we are defining now, Qpis nothing
butthe difference of whole square.
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but whatwe want is this W(ti) with W(ti+1)
- W(ti).
00:03:02.520 --> 00:03:10.360
So we started with Qpthat is equal to the
differenceof whole square.The difference of
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wholesquare is the same as suppose youtreat
this as a, this as b.
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So (a-b)2is same as a2-b2-2(a-b)-2b(a-b).So
if you simplify, you'll get (a-b)2.
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But if you expand this summationnow, this
will be the only the last termwill be there
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and all other termsvanishes.So therefore,
you will (W(T))2-(W(0))2.
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The last term and the first term willexist
and all other terms will vanishbecause of
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minus squares.
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Whereas this one, -2?_(i=0)^(n-1)¦?W(ti)?(W(ti+1)-W(ti)).
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That's what we want as n-->8.
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But we knowthe result, limitnorm of p-->0
ofQpwill be T. That we have discussed in the
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lecture 1, whenwediscussed the quadratic variation
ofBrownian motion.In the lecture 1, wehave
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discussed quadratic variation ofBrownian motion
in that Qpisdefined limit norm of p-->0 Qpwill
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be T.
Therefore, what we want is a limit n-->8 of
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this quantity and that is inthe third term
in this Qp.
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So now youapply a limit n-->8 in equation;
so the left-hand sidebecomes the T, the right-handbecomes
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(W(T))2- (W(0))2you know that W(0) is 0 for
a standard Brownian motion.
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Therefore, we will get only thefirst term;
second term will be 0, thirdterm is the unknown.That's
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the integration.
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n-->8 of summation that is nothing but the
I(T).
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Therefore, ?_0^T¦?W(t)d(t)? that is nothing
but in the left side we got T, therefore,
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we will get (W(T))2-T is equal to, since it
is -2 times, therefore it will be right.
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Hence, the integration ?_0^T¦?W(t)d(t)?
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becomes ((W(T) )^2-T)/2.
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So here we have used the quadratic variation
of Brownian motion is T that we have used
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for the the interval 0 to T.
Therefore, the integration 0 to T so we have
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used the Brownian motion quadratic variation
between the interval 0 to T that is T. Therefore,
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the integration is going to be ((W(T) )^2-T)/2.
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Hence, this integration, ((W(T) )^2-T)/2,
we see that unlike Reimann integral, we have
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extra term T. If the integration is with respect
to boundedvariation differentiable function
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thenyou won't have the term T here.
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Just it is ((W(T) )^2)/2.
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Suppose you replace W(s) by x, therefore you'll
get
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?_0^T¦?X(x)d(x)? that is nothing but x2/2
whenever the integrant is boundedvariation
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as well as differentiablefunction.That is
nothing but the Riemannintegral.So in the
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Riemann integral youwon't have the extra term
T/2, but here we have the extra term minus
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T/2.
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So this arises from the quadraticvariation
of a Brownian motion which isfinite whereas
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if it is a real valuefunction which is bounded
then thequadratic variation is going to be
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0.Buthere, the quadratic variation which isfinite
value, therefore you are gettingminus -T/2.
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So the nextremark, the above integral is defined
notonly for the upper limit integration, T,
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but also every upper limit ofintegration between
0 to T.Inthis integration, we have done the
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upperlimit as T, therefore we used the quadratic
variation of Brownian motion between the interval
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0 to T thatis T. But instead of that, we cango
for variable,t, also.
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In that case, here you have to use the quadraticvariation
of Brownian motion between theinterval 0 to
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t and that will be t itself.
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That's a variable t.
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Therefore, thisintegration will be ?_0^t¦?W(s)dW(s)?
is nothing but s is nothing but ((W(T) )^2-T)/2
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where both the ts are small variable.
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You see the construction of Ito integralis
similar to one of the Stieltjes integration,
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but instead of integratingwith respect to
the deterministicfunction theIto integral
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with respect tothe random function most precisely
pathof Weiner process.So that is adifference
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between the Ito integral andthe usual integral.The
usual integral iswith respect to the deterministicfunction
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whereas in the Ito integral, theintegration
with respect to the path ofBrownian motion
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or Weiner process whichis unbounded variation
and nowheredifferential.
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Therefore, the whole integration is different.
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Consider this stochastic integral.
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In this, first integral is a Riemannintegral
of stochastic integrant whereasthe second
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integral is aIto integral.
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This equation is very useful in finance.
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In the Black Scholemodel for pricing, the
stock price is seeming to followgeometricBrownian
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motion.
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Now we are discussing the few propertiesof
Ito integrals.Still we haveused a few properties
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in the examples, but now you can understand
how theproperties will be used in the examples.
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The following resultsholds by the Itointegral
defined in the equation 1.
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Inthe equation 1, we have discussed - yeah,
this is the Ito integral which we havediscussed
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in the first equation.So thiswe are referring.
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Yes, the first property, the integral I(t)
T is a martingale with respect to thefiltration
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F(t).
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This is a veryimportant property.If you have
a Itointegral that means the integrant isadopted
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and the mean square integraland the Ito integral
is a stochasticprocess and that stochastic
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process ishaving the martingale property withrespect
to the filtration F(t).
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The same F(t) the integrant is adoptedalso.With
respect to the same filtration, the integrant
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is adoptedalso.
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So the proof is if youwant to verify the stochastic
processes martingale you have to check threeproperties;
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the first property is it hasto be the stochastic
processfor fixed T, it has to be integrable.
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Then third property the conditionalexpectation
has to satisfy the equality property.So if
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these threeproperties are satisfied by thestochastic
process then we say thestochastic process
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has the martingale property.So here I(t) is
a stochasticprocess for t=0.
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So first, we have to verify it isintegrable.So
you can find out theexpectation of I(t) or
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every T that is going to be a finite value.Then
the second one, I(t)has to be -- for fixed
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T, I(t) has to be a F(t) measurablethat is
adopted.So for fixed alreadythe integrant
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is adopted, the integrationwith respect to
W(T) and W(T) is alsoadapted to F(t) one can
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prove, the I(t) is also adapted to the filtration
F(t).
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The third condition, expectation of I(t) given
F(s)where s less than t that is same as I(s).
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You have to prove expectation of I(t) given
F(s) wheres is less than t that is equal to
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I(s) if that is proved for all t.
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Sincethese three properties of martingale
is satisfied, I(t) will be a martingale withrespect
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to the filtration F(t).
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Here,I am not discussing the proof, here I
am not giving the proof, but I havediscussed
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whatever all the propertieshas to be verified.
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Second property, expectation of I(t) that
is same as aexpectation of 0 to t, integration
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the, definition that will be 0.
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Since it is a martingale it will have a constant
mean.
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Therefore, expectation of I(t) issame as expectation
of I(0) and youknow how to evaluate expectation
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of I(0) that will be 0.
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And the third property, expectation of a (I(t))2
that is same as expectation of ?_0^t¦?X^2(s)ds?
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that is same as ?_0^t¦?E(X^2(s))ds?So the
expectation and the integration interchange
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the place.
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This is called Ito isometry.
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So thisis a very important property of secondorder
expectation that isthe integrant and earlier
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you haveexpectation of (I(t))2 thatis nothing
but ?_0^t¦?X^2ds? this is nothing but the
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Reimann integral.
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For fixed s, the expectation of X2(s) is the
function of s.So you are integrating ?_0^t¦?E(X^2
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(s)ds?.So this is nothing but the Reimann
integral.
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Now based on theproperty number 2 and 3, you
can conclude Var(I(t)) is nothing but ?_0^t¦?E(X^2
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(s)ds?because the E(I(t))=0, therefore Var(I(t))
= E(I(t)2)-E(I(t))2 that is
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same as
?_0^t¦?E(X^2 (s)ds?.
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One canfind the quadratic variation of I(t)
also because I(t) is a stochasticprocess.So
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you can find out thequadratic variation between
the interval 0 to t.
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So [I,I] between the interval 0 to t, that
means a second ordervariation between the
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interval 0 to t for the function I(t).
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I(t) is a Itointegral whereas the quadratic
variationis nothing but integration ?_0^t¦?X^2
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(s)ds?.So this is the Reimann integral.
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So I have now given the proof ofthe fifth
property also.We are juststating the results
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of Ito integral andwe have used a few properties
in theexamples.
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The Ito integral is a random variablefor all
T, for all W belonging to O. Therefor,e you
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can write down, I(t)(w) that is nothing but
limit n-->8, the value X(si), the difference
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of Ws.
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You know that X is a random variable, thedifference
of Ws is also a randomvariable.Therefore,
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for fixed n, thissummation is equal to 0 to
n-1 will be nothing but a random variable.
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Weare finding limit n-->8 of this summation.Therefore,actually
theconvergence takes place only for thesubsequence.That
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means I(t) is arandom variable and that random
variableis nothing but the convergence of
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theright-hand side, limit n-->8 of this summation
and itsintegral depends on the sample pathbecause
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you are finding the differenceof W(si+1)-W(si)
therefore, this integral depends onthe sample
00:19:08.450 --> 00:19:09.970
path.
00:19:09.970 --> 00:19:19.660
Also, we need these following properties the
dW(t)dW(t) will be dt that is nothing but
00:19:19.660 --> 00:19:26.160
the quadratic variation of Brownian motionis
nothing, but a finite value if you arefinding
00:19:26.160 --> 00:19:33.940
the increment between theinterval 0 to t then
the quadraticvariation is small t, whereas
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if it is areal value function, which is adifferentiable,
then dtdt, the quadraticvariation of the function
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t will be 0.
00:19:46.590 --> 00:19:51.950
Andthe mixed variation, cross-variation dW(t)dt
that will be 0.
00:19:51.950 --> 00:20:00.051
So if you find out the quadratic variation
withthe W(t) that will be the t, whereas a
00:20:00.051 --> 00:20:06.550
quadraticvariation with the function t will
be 0and the cross-variation will be 0.Sothis
00:20:06.550 --> 00:20:17.860
result will be used when you arefinding the
I(t).
00:20:17.860 --> 00:20:26.740
Here are theimportant referencesfor the Ito
integrals.In the next class, we will consider
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the Ito formulas.