The Itō integral offers a way to integrate
a given stochastic processGt
with respect to a Wiener processBt,
which is also a stochastic process.
The Itō integral It of Gt is defined as follows:
Where have partitioned the time interval [a,b] into steps of size h.
The above integral exists if Gt and Bt are adapted
to a common filtration Ft,
and E[Gt2] is integrable for t∈[a,b].
If It exists, Gt is said to be Itō-integrable with respect to Bt.
Consider the following simple first-order differential equation for Xt,
for some function f:
This can be solved numerically using the explicit Euler scheme
by discretizing it with step size h,
which can be applied recursively, leading to:
In the limit h→0, this leads to the following unsurprising integral for Xt:
In contrast, consider the stochastic differential equation below,
where ξt represents white noise,
which is informally the t-derivative
of the Wiener process ξt=dBt/dt:
Now Xt is not deterministic,
since ξt is derived from a random variable Bt.
If g=1, we expect Xt=X0+Bt.
With this in mind, we introduce the Euler-Maruyama scheme:
We would like to turn this into an integral for Xt, as we did above.
Therefore, we state:
This integral is defined as below,
analogously to the first, but with h replaced by
the increment Bt+h−Bt of a Wiener process.
This is an Itō integral:
For more information about applying the Itō integral in this way,
see the Itō calculus.
Since Gt and Bt must be known (i.e. Ft-adapted)
in order to evaluate the Itō integral It at any given t,
it logically follows that It is also Ft-adapted.
Because the Itō integral is defined as the limit of a sum of linear terms,
it inherits this linearity.
Consider two Itō-integrable processes Gt and Ht,
and two constants v,w∈R:
By adding multiple summations,
the Itō integral clearly satisfies, for a<b<c:
A more interesting property is the Itō isometry,
which expresses the expectation of the square of an Itō integral of Gt
as a simpler “ordinary” integral of the expectation of Gt2
(which exists by the definition of Itō-integrability):
We write out the left-hand side of the Itō isometry,
where eventually h→0:
Recall that Gt and Bt are adapted to Ft:
at time t, we have information Ft,
which includes knowledge of the realized values Gt and Bt.
Since t≥s+h by assumption, we can simply factor out the known quantities:
For the existence of It,
we need E[Gt2] to be integrable over the target interval,
so from the Itō isometry we have E[I]2<∞,
and therefore E[I]<∞,
so It has all the properties of a Martingale,
since it is trivially Ft-adapted.
Lecture notes on diffusions and stochastic differential equations,
2021, Polyteknisk Kompendie.