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# 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 = t0 < t1 < t2 < — < tn _ j < tn = d and approximate x values w0 < Wj < w2 < — < wn to be determined at the respective 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 Xc 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 {w0, w1, ..., wn} is an approximation realization of the stochastic solutionX(t), which depends on the random number z{ that was chosen. Since Wt is a stochastic process, each realization will be different and so will our approximations.

## Milstein Method

The approximation scheme of the Milstein method for Equation 3.32 may be written in the following way (Sauer 2012):  Found a mistake? Please highlight the word and press Shift + Enter Subjects