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:
It≡∫abGtdBt≡h→0limt=a∑t=bGt(Bt+h−Bt)
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.
Motivation
Consider the following simple first-order differential equation for Xt,
for some function f:
dtdXt=f(Xt)
This can be solved numerically using the explicit Euler scheme
by discretizing it with step size h,
which can be applied recursively, leading to:
Xt+h≈Xt+f(Xt)h⟹Xt≈X0+s=0∑s=tf(Xs)h
In the limit h→0, this leads to the following unsurprising integral for Xt:
∫0tf(Xs)ds=h→0lims=0∑s=tf(Xs)h
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:
dtdXt=g(Xt)ξt
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:
Xt=X0+∫0tg(Xs)dBs
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:
∫0tg(Xs)dBs≡h→0lims=0∑s=tg(Xs)(Bs+h−Bs)
For more information about applying the Itō integral in this way,
see the Itō calculus.
Properties
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:
∫abvGt+wHtdBt=v∫abGtdBt+w∫abHtdBt
By adding multiple summations,
the Itō integral clearly satisfies, for a<b<c:
∫acGtdBt=∫abGtdBt+∫bcGtdBt
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):
E(∫abGtdBt)2=∫abE[Gt2]dt
We write out the left-hand side of the Itō isometry,
where eventually h→0:
In the particular case t≥s+h,
a given term of this summation can be rewritten
as follows using the law of total expectation
(see conditional expectation):
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:
Furthermore, Itō integrals are martingales,
meaning that the average noise contribution is zero,
which makes intuitive sense,
since true white noise cannot be biased.
We will prove that an arbitrary Itō integral It is a martingale.
Using additivity, we know that the increment It−Is
is as follows, given information Fs:
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.
References
U.H. Thygesen,
Lecture notes on diffusions and stochastic differential equations,
2021, Polyteknisk Kompendie.