# Euler-Maruyama and Milstein Methods

Numerical methods for the SDEs are well known. Accordingly, one may consider the Euler-Maruyama method, which is used to solve the said uncertain problems. As such, let us assign a grid of points, *c* = t_{0} < *t _{1}* < t

_{2}< — <

*t*_ j <

_{n}*t*=

_{n}*d*and approximate x values

*w*< Wj <

_{0}*w*< — <

_{2}*w*to be determined at the respective

_{n}*t*points.

Let us consider SDE initial value problem (Black and Scholes 1973)

As said earlier, numerical schemes for Equation 3.31 have been incorporated for the two well-known methods (i.e. Euler-Maruyama and Milstein), which is discussed next.

## Euler-Maruyama Method

We take a time-discrete approximation of the SDE (Higham 2001)

Then the approximation scheme for Euler-Maruyama may be represented as follows (Sauer 2012):

where *X _{c}* is the value of

*X*at

*t*= c,

We define *N(0,* 1) to be the normal distribution
and each random number Д is computed as

The obtained set *{w _{0}, w_{1},* ...,

*w*is an approximation realization of the stochastic solutionX(t), which depends on the random number

_{n}}*z*that was chosen. Since

_{{}*W*is a stochastic process, each realization will be different and so will our approximations.

_{t}## Milstein Method

The approximation scheme of the Milstein method for Equation 3.32 may be written in the following way (Sauer 2012):