Absorbing floor and terminal-wealth distribution

Probability decomposition

At horizon \(T\), total probability is split into:

  1. survived wealth above the floor;

  2. an atom at the floor from diffusion paths that hit the absorbing barrier; and

  3. wealth below the floor from crash jumps that overshoot the barrier.

The three masses should reconcile to one within numerical integration tolerance.

Public functions

Function

Output

compute_survival

probability of remaining above the absorbing barrier

compute_density

regime-conditional bounded transition densities

compute_tilted_survival

exponentially tilted survival transform used in moments

compute_overshoot_density

density of jump distance beyond the barrier

For a terminal gap process, integrate compute_density on the positive gap grid and compute_overshoot_density on the overshoot grid. The floor atom is the residual \(1-S-O\) after survival mass \(S\) and overshoot mass \(O\) are computed. Use a grid that is wide and fine enough for the desired tolerance; do not clip a negative residual silently without checking integration error.

Units and interpretation

  • Density grids are in the log-gap or overshoot coordinate expected by the corresponding API, not directly in wealth units.

  • Wealth mapping requires the target wealth and buffer from the same Riccati solution.

  • A positive overshoot mass is expected in a jump model and is economically different from the diffusion floor atom.

Runnable source

examples/terminal_wealth_distribution.py constructs the full decomposition and overlays an independent Monte Carlo histogram. It saves one PNG. For regression tolerances, see validation.

Failure modes

  • Root-finding failure when Laplace characteristic roots are not separated cleanly.

  • An inversion/grid range that truncates material tail mass.

  • Mixing an asset specification with a gap process produced by another Riccati solution.

  • Confusing the exponential jump rate with its reciprocal mean.

API: Laplace and regime-switching modules.