[x] I would change the notation a little. Instead of (126) and (127) having the fudge on the f(x) built in, you should instead have it additive. i.e.
[L B] v_bar = [f b]
becomes
L^B v = f + b^B
where b^B is 0 if homogenous and in that example
b^B = [Delta^-1 * S; 0;. ....]
The basic goal of this function is to generate the b^B function. (Resolved by fd4c218)
[x] In the math, expand (205) to include the various versions of this for L_1+, L_1-, L_2, and L_0 for absorbing boundaries at the top and bottom -- work on irregular cases first
[x] L_1+
[x] L_1-
[x] Maybe give an example of how it composes as well when you have a sequence of L_1, L_2, etc. (Resolved by 6c681c1)
[x] I think the derivations of those can become quite simple... However, we will ultimately want to look at the composition? i.e.
L = r - mu L_{1-} + sigma^2/2 L_2
etc. That way you can pass in the composed operator to get the b^B.
If we only support relatively simple examples (e.g. non-homogenous absorbing barriers at one or both sides) then is there a formlua we could apply to do this? Do we need to pass in the composed extension operator, for example?
The discretization of the differential operators and boundary conditions all assume homogeneity of the BCs to be able to use blindly. I found (128) and (205) as an examples of where this algebra is done for a particular boundary condition, and I found the formulas in in https://quantecon.github.io/SimpleDifferentialOperators.jl/dev/examples/#Solving-HJBE-with-absorbing-barrier-conditions-1 but I am not sure if I see the general approach in that setup.
f(x)built in, you should instead have it additive. i.e.becomes
where
b^Bis0if homogenous and in that exampleThe basic goal of this function is to generate the
b^Bfunction. (Resolved by fd4c218)etc. That way you can pass in the composed operator to get the
b^B.If we only support relatively simple examples (e.g. non-homogenous absorbing barriers at one or both sides) then is there a formlua we could apply to do this? Do we need to pass in the composed extension operator, for example?