Solve steady state KFE with absorbing barrier and entry/exit

[jlperla]

We can add this to a new KFE notebook for looking at evolutions of KFE with birth-death.

The setup will be a:

• Unit mass of agents.
• Random death rate of delta >= 0 each period
• Absorbing barrier at x=0 and a reflecting barrier at some x_max
• constant coefficient brownian motion evolution in the interior
• New entrants of the exact same mass as those crossing the absorbing barrier/dying, and enter along an exogenous distribution h(x) with support on x = [0, x_max].

In differentiable form, I believe the setup is then

D_t f(t,x) = - mu D_x f(t,x) + sigma^2/2 D_xx f(t,x) - delta f(t,x) + S(t) h(x)
f(t, x) = 0 for all x <= 0
- 2 mu/sigma^2 f(x_max, t) + D_x f(x_max, t) = 0
S(t) = delta + the flux at 0 boundary 

where int_0^{xmax} h(x) dx = 1

• [ ] Figure out the formula for the flux across the boundary for the arbitrary f(t,x) function
  • I think it is S(t) = delta + mu f(t, 0) + sigma^2/2 D_x f(t,0) but you should verify.
• [ ] Double-check that the KFE with the insertion of agents and random death with distribution h(x) above is correct.
• [ ] Figure out the right way to discretize the h(x) to ensure it is conservative. i.e. pdf at all interior points renormalized to sum to 1, etc.?
• [ ] See if there are ways to make it conservative without making it nonlinear in the system of equations?

[jlperla]

For an h(x) example, you should try a few things. First, do an impulse by choosing some x in the grid at node m and then h = [0; 0;. .... 1 ; 0;. 0] etc. i.e. only have a value at the m'th point.

Also try a truncated exponential, which will end up being the main one we are interested in.

The continous equation for an exponential , truncated at xmax and with parameter theta is

θ = 2.0
x̄ = 4.0
f(x) = θ * exp(-θ * x) / (1 - exp(-θ * x̄))

To discretize on a grid

# Discretize on grid x?
x = range(0.0, x̄, length = 10)
f̂ = f.(x) / sum(f.(x) )

# Or should we be taking out the ends?
f̂ = f.(x[2:end-1]) / sum(f.(x[2:end-1]) )