Let us consider a standard stochastic differential equation:
which is written in differential form.
The integral form of Equation 11.1 becomes
where the last term on the right-hand side of Equation 11.2 is called the Ito integral.
We take c = t0< t1< t2< ••• < tn_ 1 < tn = d as a grid of points on an interval [c, d], then the Ito integral may be defined in the following limit form:
where AW, = Wti - Щ-и a step of Brownian motion across the interval.
Analytical Solution of Stochastic Differential Equations
First, Equation 11.1 is solved analytically by using the Ito formula. If Xt is an Ito process, then we have
Let g(t, x) e C2([0, те] x R) (i.e. g is twice continuous differentiable on [0, те] x R). Then Yt = g(t, Xt) is again an Ito process (Oksendal 2003) and one can write
where (dXt)2 = (dXt) • (dXt), which is computed as follows:
Let us consider an SDE (Malinowski and Michta 2011):
where at = rt + aWt
Wt and a are noise and constant, respectively Equation 11.6 may be written as
Using the Ito formula for the function g(t, x) = ln x (Oksendal 2003), we get the following: Integrating this, we get
It may be noted from the literature that except for some standard problems, the exact method may not be applicable for others. Hence, we need a numerical treatment to handle non-trivial problems, and this has been discussed in the following sections.