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):

 
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