Numerical Benchmarks on the Stochastic Maltus Model

We will compare Euler-Murayama and Milstein methods to solve the stochastic Maltus model for various possible values of the parameters. Recall Maltus stochastic model:

$dx_t = r x_t dt + c x_t dw_t.$

As we have shown, its analytic solution is

$x_t = x_0 \exp\left(\left(r - \frac{c^2}{2}\right) t + c w_t.\right)$

We are going to implement Euler-Murayama’s and Milstein’s in numpy and julia. Numpy implementation can be found in the notebook used to perform the calculations. Likewise, julia implementation can be downloaded here.