Currently I'm learning about SDE's. For an assignment I need do realise tracks of the Black-Scholes equation. The realisation of these tracks is not the problem.
I need to show that the rate of convergence in the weak sense is dt and in the strong sense dt^0.5. In my notes, I read that in order to illustrate the weak sense you need to realise many tracks so that the dt^0.5 cancel out.
The problem I run into is that I don't know how to calculate the global error for each time-step. I want to create a figure like the one attached.
You can see on the y-axis the global error on T =10. I have the following formula for the Black scholes equation:
$$\newcommand{\D}{\mathit{\Delta}} \begin{align} X_{t+\D t}&=X_t+(α-\tfrac12σ_t^2)\D t+σ_tz^1_t\sqrt{\D t}\\ \end{align} $$
Imagine that I realise 10000 tracks with this equation. Then I go to the place in the array where the value for X=10 is stored. What do I need to in order to obtain the global error for that dt?
Thanks, Tim
P.S
In the my course notes the following was written, but I don't know how this is helpful to empirically show the rate of convergence.
$$ \begin{array}{l} E_{N}=E_{N-1}+O\left(\Delta t^{2}\right) \\ E_{N-2}+2 * O\left(\Delta t^{2}\right)= \\ N^{*} O\left(\Delta t^{2}\right)= \\ T * O(\Delta t)=O(\Delta t) \end{array} $$
