Where the latter is an Itō integral,
assuming Gt is Itō-integrable.
We call Xt an Itō process if Ft is locally integrable,
and the initial condition X0 is known,
i.e. X0 is F0-measurable,
where Ft is the filtration
to which Ft, Gt and Bt are adapted.
The above definition of Xt is often abbreviated as follows,
where X0 is implicit:
Typically, Ft is referred to as the drift of Xt,
and Gt as its intensity.
Because the Itō integral of Gt is a
it does not contribute to the mean of Xt:
Now, consider the following Itō stochastic differential equation (SDE),
where ξt=dBt/dt is white noise,
informally treated as the t-derivative of Bt:
An Itō process Xt is said to satisfy this equation
if f(Xt,t)=Ft and g(Xt,t)=Gt,
in which case Xt is also called an Itō diffusion.
All Itō diffusions are Markov processes,
since only the current value of Xt determines the future,
and Bt is also a Markov process.
Classically, given y≡h(x(t),t),
the chain rule of differentiation states that:
However, for a stochastic process Yt≡h(Xt,t),
where Xt is an Itō process,
the chain rule is modified to the following,
known as Itō’s lemma:
In the limit of small dt, we can neglect dt2,
and as it turns out, dtdBt too:
However, due to the scaling property of Bt,
we cannot ignore dBt2, which has order dt:
Where χ12(dt) is the generalized chi-squared distribution
with one term of variance dt.
The most important application of Itō’s lemma
is to perform coordinate transformations,
to make the solution of a given Itō SDE easier.
The simplest coordinate transformation is a scaling of the time axis.
Defining s≡αt, the goal is to keep the Itō process.
We know how to scale Bt, be setting Ws≡αBs/α.
Let Ys≡Xt be the new variable on the rescaled axis, then:
It is worth knowing under what condition a solution to a given SDE exists,
in the sense that it is finite on the entire time axis.
Suppose the drift f and intensity g satisfy these inequalities,
for some known constant K and for all x:
When this is satisfied, we can find the following upper bound
on an Itō process Xt,
which clearly implies that Xt is finite for all t:
If we define Yt≡Xt2,
then Itō’s lemma tells us that the following holds:
Integrating and taking the expectation value
removes the Wiener term, leaving:
Given that K(1+x2) is an upper bound of xf(x) and g2(x),
we get an inequality:
Where we have implicitly used that DsFs=∣DsFs∣
because Zt is positive for all Gs2,
and that ∣Ds∣2=Ds2 because Ds is real.
We then apply the
recognizing that Z0 does not decrease with time (since it is constant):
Using these properties, it can then be shown
that if all of the above conditions are satisfied,
then the SDE has a unique solution,
which is Ft-adapted, continuous, and exists for all times.
Lecture notes on diffusions and stochastic differential equations,
2021, Polyteknisk Kompendie.