Killed Brownian motion with a prescribed lifetime distribution and models of default

The inverse first passage time problem asks whether, for a Brownian motion B and a nonnegative random variable ., there exists a time-varying barrier b such that P{Bs>b(s),0.s.t}=P{.>t}. We study a .smoothed. version of this problem and ask whether there is a .barrier. b such that E[exp(...t0.(Bs.b(s))ds)]=P{.>t}, where . is a killing rate parameter, and .:R.[0,1] is a nonincreasing function. We prove that if . is suitably smooth, the function t.P{.>t} is twice continuously differentiable, and the condition 0<.dlogP{.>t}dt<. holds for the hazard rate of ., 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.