 # Preliminary

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 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: Example 11.1

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. 