"Gentle Introduction to Upwind Finite Differences"

[jlperla]

As discussed, @chiyahn will be the "project manager" on helping to get the expositional notebook done that we talked about today and @ajozefiak can implement stuff as you guys work through stuff. A few of the things we discussed:

• Maybe call it "Gentle Introduction to Upwind Finite Differences"?
• The motivation for the example is just valuing an asset with a stream of payoffs. Basically, understanding the entirely of https://quantecon.github.io/SimpleDifferentialOperators.jl/stable/#Solving-HJBE-with-constant-drifts-1 code and what different steps are doing.
• Call it "Bellman" equations and motivate by just pricing of assets and payoffs rather than using the term HJBE. I am a little concerned people will get confused if there isn't a fancy hamiltonian part.
• Stick to the simple case of constant drift and constant variance with a negative drift through almost all of it, and maybe do the extension to solve the non-constant drift with a sign change at the bottom as an extension
• Make sure you show the basic approaches to discretizing the sub-components of the operators and how to construct the matrices... then show the SDO.jl code that does it
• Try to mix up code and math however you can.
• It may be better to show how to solve with the extension operators and solving the system first, and then look at how the boundary conditions can be applied to the operators themselves after.

As for the code, follow the lead and notation on https://github.com/QuantEcon/SimpleDifferentialOperators.jl/blob/master/docs/examples/LCP_simple.jmd and the docs wherever possible. The main example code would be to describe https://quantecon.github.io/SimpleDifferentialOperators.jl/stable/#Solving-HJBE-with-constant-drifts-1 and consistency will be important. If you change notation in one, get it consistent.

At the bottom where you deal with state-dependent drifs, have the code consistent with https://quantecon.github.io/SimpleDifferentialOperators.jl/stable/#Solving-HJBE-with-state-dependent-drifts-1

[chiyahn]

Instead of state dependent drift, focus on constant drift at this moment (don't worry about upwind schemes)

• [ ] Explain how an ODE can be represented in a matrix form
  • [ ] After discretizing the states, show how derivatives can be represented in FD methods, and form a linear system
  • [ ] And form a discretized differential operator (with code)
  • [ ] And show we can use SDO package for this

For barebone (due this Tuesday/Wednesday) Keep in mind that you should now pass extended grids to differential operators -- try working with the current dev version with ] dev https://github.com/QuantEcon/SimpleDifferentialOperators.jl/

• [ ] Symbols x_bar, x, L for linear operators.. L_x
• [ ] M = 3 case and explain how L_1-/L_1+ and L_2 can be constructed
• [ ] Point out we are using L_1- for negative drift
• [ ] Working code

(CY)

• [ ] Extended operator, explain how ghost nodes can be added
• [ ] Show extended system, solve it
• [ ] Show how SDO can be used
• [ ] Review