The inverse first passage time problem asks whether, for a Brownian motion B and a nonnegative random variable \zeta, there exists a time-varying barrier b such that P\{B_s>b(s), \, 0 %DIFDELCMD < \le %%% s%DIFDELCMD < \le %%%  t\}=P\{\zeta>t\}. We study a "smoothed" version of this problem and ask whether there is a "barrier" b such that \mathbb{E[\exp(-\lambda \int_0^t \psi(B_s - b(s)) \, ds)] } [\exp(-\lambda\int_0^t\psi(B_s-b(s))\,ds)]} =P\{\zeta >t\}, where \lambda is a killing rate parameter  and \psi:R %DIFDELCMD < \to [%%% 0,1 is a non-increasing \to[0,1]  function. We prove that if \psi is suitably smooth, the function t\mapsto P\{\zeta>t\} is twice continuously differentiable, and the condition 0<- \frac{d \log \mathbb{P\{\zeta > t\}}{dt} } \{\zeta>t\}}{dt}} <\lambda holds for the hazard rate of \zeta, 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.
The inverse first passage time problem asks whether, for a Brownian motion B and a nonnegative random variable \zeta, there exists a time-varying barrier b such that P\{B_s>b(s), 0 %DIFDELCMD < \le %%% %DIFDELCMD < \le %%% \leq s\leq t\}=P\{\zeta>t\}. We study a "smoothed" version of this problem and ask whether there is a "barrier" b such that [\exp(-\lambda \int_0^t \psi(B_s - b(s)) \, ds)] } \mathbb{E[\exp(-\lambda\int_0^t\psi(B_s-b(s))\,ds)]} =P\{\zeta >t\}, where \lambda is a killing rate parameter , and \psi:R %DIFDELCMD < \to [%%% \to[0,1] is a nonincreasing function. We prove that if \psi is suitably smooth, the function t\mapsto P\{\zeta>t\} is twice continuously differentiable, and the condition 0<- \{\zeta > t\}}{dt} } \frac{d\log\mathbb{P\{\zeta>t\}}{dt}} <\lambda holds for the hazard rate of \zeta, 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.
D
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D
s
A
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\mathbb{E
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\mathbb{E
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0,1
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is a non-increasing
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is a nonincreasing
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\frac{d \log \mathbb{P
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\frac{d\log\mathbb{P