Killed Brownian Motion with a Prescribed Lifetime Distribution and Models of Default

Abstract. The inverse first passage time problem asks whether, for a Brownian motion B and a nonnegative random variable ζ, there exists a timevarying barrier b such that P{Bs > b(s), 0 ≤ s ≤ t} = P{ζ > t}. We study a “smoothed” version of this problem and ask whether there is a “barrier” b such that E[exp(−λ ∫ t 0 ψ(Bs − b(s)) ds)] = P{ζ > t}, where λ is a killing rate parameter and ψ : R → [0, 1] is a non-increasing function. We prove that if ψ is suitably smooth, the function t 7→ P{ζ > t} is twice continuously differentiable, and the condition 0 < − d log P{ζ>t} dt < λ holds for the hazard rate of ζ, then there exists a unique continuously differentiable function b solving the smoothed problem. We show how this result leads to flexible models of default for which it is possible to compute expected values of contingent claims.