Cramér-Lundberg Insurance Ruin Theory

The surplus process U(t) = u + ct − S(t) where S(t) is compound Poisson claims. Ruin occurs when U(t) hits zero. The Cramér-Lundberg theorem gives ψ(u) ≤ e^(−Ru) where R is the adjustment coefficient.

Controls

Net profit θ = c/(λμ)−1: —
Empirical ruin prob: —
Theoretical ψ(u) ≤: —
Ruined paths: —
U(t) = u + ct − S(t)
S(t) = Σ Xᵢ (compound Poisson)

Net profit: θ = c/(λμ) − 1 > 0
R = adjustment coefficient
ψ(u) ≤ exp(−Ru)

Exact (exp claims):
ψ(u) = (λμ/c)·exp(−Ru)
R = 1/μ − λ/c