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 = t_{0}< t_{1}< t_{2}< ••• < t_{n}_ _{1} < t_{n} = 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, = W_{ti} - 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 X_{t} is an Ito process, then we have

Let g(t, x) e C^{2}([0, те] x R) (i.e. g is twice continuous differentiable on [0, те] x R). Then Y_{t} = g(t, X_{t}) is again an Ito process (Oksendal 2003) and one can write

where (dX_{t})^{2} = (dX_{t}) • (dX_{t}), which is computed as follows:

Example 11.1

Let us consider an SDE (Malinowski and Michta 2011):

where a_{t} = r_{t} + aW_{t}

W_{t} 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.