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Language: en
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This
is Stochastic Processes,Module 7, Brownian
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Motion and its Applications.
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Lecture 4, Ito Calculus Ito integrals.
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In the last three lectures we havediscussed
the definition andpropertiesof Brownian motion
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in the first lecture.
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In the second lecture, we have discussed
Geometric Brownian motion and the otherprocess
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derived from the Brownianmotion, and their
properties also.Thethird lecture we have discussedstochastic
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differential equations.In
this lecture we are going to discuss Ito integrals.
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First, we are going to startwith the definition
of Ito integral.
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Then we say the stochastic process is this
Ito integral.
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Then we are going todiscuss what is Ito process.Then
followed
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by these definitions,we are going todiscuss
few examples.Followed by theexamples we are
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going to discuss theproperties of Ito integrals.So
with thatthis lecture will be completed.
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This isthe definition of Ito integral.
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Let X(t) bea stochastic process which is adapted
tothe natural filtration F(t) of a Wiener
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process W(t).
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That is X(t) be a F(t) measurable.Define
I(t) that is nothing butintegration between
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the limit 0 to t X(u) integration with respect
to W(u).
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That t lies between 0 to T where t is a positive
constant.
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I(t) be a stochastic integral with respect
to a Wiener process.
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The above integralis called the Ito integral.
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A detailedthe interpretation and the motivation
ofthis can be found in the reference books.
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So the Ito integral I(t) is definedin the
form of integration 0 to t andthe integrant
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with respect to theintegral with respect to
the Wiener process, W(t).The Wiener processalready
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discussed, the Wiener processor Brownian motion
is discussed in Lecture 1, you can find in
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Lecture 1, what are allthe properties of Wiener
process andso on.The Wiener process W(t) isstarting
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with the standard one, W0 0 to 0 and it has
increments of
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stationary as well as independent andthe increments
are normally distributedwith the mean 0 or
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the standard Wiener process and the variants
is an increment, the difference,for s is less
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than t, the variance is t-s.
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So here we have a stochastic process, X(t)
which is defined between the interval 0to
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T.The same stochastic processis adopted to
the natural filtration, F(t).
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Already we know that the Wiener process is
adopted to the W(t) and here X(t) is also
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adopted to the W(t)that meansfor fixed t,
X(t) is a F(t) measure.
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That means in the Sigma field generatedby
the X(t) for fixed t, will becontained in
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F(t).
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That means theinformation accumulated at time
T thatis sufficient to find out the value
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of X(t) for fixed t.
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So the X(t) is F(t) measurable for all t between
the interval, 0 to T.
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Then we can define thisintegral will be call
it as aItointegral.So the X(t) has to be F(t)
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measurable and F(t) is a naturalfiltration
for the Wiener process, W(t), then for all
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values of T between theinterval 0 to T, I(t)
iscalled stochastic integral because this
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is not the issued integral; this is a stochastic
integral with respect to the Weiner process,
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W(t).
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This is the definition of Ito integral.
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Now we are going to discuss the Ito process.
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Let W(t) be the Brownian motion and F(t) be
the associated natural filtration.
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A Ito process is astochastic process, X(t),
is of the form
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X(t)=X(0)+ ?_0^t¦?(u)dW(u) + ?_0^t¦?(u)du
So here wehave two types of integration; the
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onetype is the usual Riemann integration,
the other one is the Ito integral where X(0)is
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a non-random and ?(u) as well as?(u)are adopted
process and the?(u) is a mean square integral.If
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this condition is satisfied then we cansay
this X(t) is going to be aIto process.For
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all T, we are able to write X(t) is of the
form the one Ito integral andthe Riemann integral
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and both theintegrals are adopted process,
that means
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X(u) is a F(u) measurable as well as?(u) is
F(u) measurable as well as ?(u) is a mean
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square
Integral, that means that?_0^t¦?(u) du is
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a finite one.
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Then we say it is mean square integral.If
these threeconditions are satisfied by the
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X0?(u)?(u) we can say the X(t) is going to
be an Ito process.
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Once it is aIto process, you can write down
in a differentshape form that is stochasticdifferential
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equation because we have dW(u).
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In the differential form, itwill be because
X(t) is equal to X(0) plus integration, therefore
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in a stochastic differentialform, it is
dX(t)= ?(t)dW(t)+ ?(t)dt.So thisis a stochastic
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differential form of Itoprocess.Itmeans whenever
?(t)as well as ?(t), both are adopted processand
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X(0) is non-random as well as?(t) mean square
integral then the stochastic process written
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in thestochastic differential form dX(t)=
?(t)dW(t)+ ?(t)dtwill be called as a Ito process.Sothe
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Ito process is nothing but astochastic process
is of this form aswell as satisfying these
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conditions.
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Theway use you write the stochasticdifferential
equation dX(t)= ?(t)dW(t)+ ?(t)dt
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Therefore, if you see the sample path, it
is going to be the continuousfunction.Therefore,
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all the stochasticprocess with no jumps are
actually a Ito process.Where you see the Ito
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process,
you won't find the jumps.
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A jump Ito process is nothing but a Itoprocess
in which the moments arediscrete rather than
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continuous So allthe stochastic process with
no jumps are actually a Itoprocess because
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it has the term ofdtas well as dW(t) terms
for theincrement of DX(t) for all T.
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Ito integrals as well as the Brownian motionare
the examples of Ito process.The waythey previously
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written, I(t) as well as the Brownian motion,
both arecalled the Ito processes.
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Now, we are going to discuss the Ito integral.If
X(t) and W(t) are not stochastic process,
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but
rather deterministic functions then thesituation
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is different.
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Assuming f(s) that means wediscussedthe deterministic
situation, we written ?_0^t¦g(s)df(s) where
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f(s) is a differentialfunction and g(s) is
the smoothfunction.Since it is a differentialfunction
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you can write df(s) as f'(s)ds which is nothingbut
the Riemann integral.
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Whenever X(t) and W(t) are non-stochastic
processesrather than deterministic functions
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then
you can write down this integration g(s)df(s)
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that is nothing but the Riemann integrals.
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But in our situation the f(s) is not differential.We
canstill define the integral, remember thatW(t)
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which is nowhere differentiableas well as
unbounded variation.So herewe are going to
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discuss when f(s) is not differentiable as
well as when f(s) is bounded variation then
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stillthe integral is well-defined.
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Further if the function g(s) isextremely fluctuating
at differentpoints in time, the limits mean
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stilldiverge.When f(s) is bounded variationwe
can prove that the integralis well-defined.
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Further, if the function g(s) is extremely
fluctuating at different points in time the
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limit may
still diverge as s tends to infinity.
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Since f(s) is bounded variation, onecan prove
that the limit exist as longas g(s) is not
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varying too muchand it is given by the ?_0^t¦g(s)df(s)
is nothing but
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lim?n?8?_(i=0)^(n-1)¦?g(si)(f(si+1)-f(si))?.
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Thatmeans since the integration is withrespect
to since df(s) so you can findthe difference,
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f(s+i)-f(si) then multiply it with g(si).
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That means you are partitioning theinterval
0 to t into n parts then as n tends to infinity,
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then that summation limit, ntends to infinity,
will be theintegration.
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Since f(s) is bounded variation, you can prove
thatthe limit exists.Remember that in the
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integral 1 that is this isintegration,Ito
integral I(t) X(u), X(u), X(u) is a random
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variable for
fixed u, W is a random variable for fixed
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uand W(u) is nowhere differentiable as well
a unbounded variation.
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Now wehave discussed what is the situation
if f(s) is a bounded variation, the integration
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with respect to f(s) as far as f(s) is boundedvariation.
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Whereas our Ito integral X(t) andW(t) are
stochastic processes as wellas W(t) is unbounded
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variation and nowhere differential.So that's
the differencebetween the Riemann integral
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and theintegral which we are discussing nowthat
is Ito integral.
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The integrity is X(t), the integration with
respect to W(t) where W(t) is an unbounded
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variation as well as nowhere differential
whereas the Riemannintegral which we have
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discussed, the f(s) is differentiable as well
as laterwe discuss f(s) is a bounded variation,
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whereas the Ito integral W(t) is unboundedvariation
as well far as W(t) is nowhere differential.
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Hence, it is different from the Reimann integral.
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So with the Reimann integral, the integration
with respect to thefunction which is a bounded
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variationand it is differential whereas here,
the Ito integral, it is unbounded variationas
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well as nowhere differential.
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Now you canrewrite the Ito integral 1 in theabove
form, in the following form.
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I(t) that is nothing but the way wehave written
in the above form, Riemannintegral
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?_0^t¦g(s)df(s) is limit n?8, the same way
you can write
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lim-(n?8)???_(i=0)^(n-1)¦?X(Si)(W(Si+1)-W(Si))?^
?
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The way we have written the Riemannintegral,
in the same way we are writingthe Ito integral
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where X(t) is an adoptedprocess and the W(t)
is a Weiner process or Brownianmotion and
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X(t) is adapted to the f(t) and the Ito integral
I(t) can be written as lim-(n?8) the summation
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form.