# 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 = t _{0}< t_{1}< t_{2}< ••• < t*

_{n}_

*as a grid of points on an interval*

_{1}< t_{n}= d*[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*is again an Ito process (Oksendal 2003) and one can write

_{t})

where (dX_{t})^{2} = *(dX _{t})* •

*(dX*which is computed as follows:

_{t}),

**Example 11.1**

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

where *a _{t}* =

*r*+ aW

_{t}_{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.