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