How do the boundary values change if the upwind scheme changes (i.e. sign of $\mu$)

[jlperla]

While the code is written for generic upwind and downwind processes, the baseline has the same sign for all $x$.

The question is:

• [x] what exactly are the boundary values and how do they manifest into A and how are they interpreted in the case of $\mu > 0$ vs. the one we have been focusing on with $\mu < 0$. What changes and why?

For example, in the $\mu < 0$ the complication at the right hand side was that for a maximum of $N$ points, it was trying to reference the $N+1$ due to the 2nd derivative. The first derivative was upwind and using backward differences so didn't reverence the $N+1$ directly.

• [x] Is it the same story with a $\mu > 0$? Does it use EXACTLY the same assumptions (but now working forward? We need to write this down explicitly.

• [x] Similar complications occur on the left hand boundary when done carefully.

• [x] For all cases, write things up very explicitly in optimal_stopping_notes

[jlperla]

• [x] Another part of this is what the simpest form of the $A$ matrix construction looks like if the diffusion is always backwards. We might have this in one of the forms of our model, and removing the max/min functions for the upwind could help a great deal.
• [x] Write the construction up in latex and write up the code that just verifies the construction.

[jlperla]

Most of this was completed. Closing and separate verification later.