具有指定寿命分布的Killed布朗运动与违约模型
Killed Brownian motion with a prescribed lifetime distribution and models of default
摘要：
第一通过时间反问题是关于一个布朗运动$B$和一个非负随机变量$ζ$，是否存在一个时变障碍$b$使得$\mathbb{P}\{B_s>b(s),0\leq s\leq t\}=\mathbb{P}\{ζ>t\}$ .我们研究这个问题的“平滑”版本，并询问是否有一个“障碍”$b$，使$ \mathbb{E}[\exp(-λ\int_0^tψ(B_s-b(s))\,ds)]=\mathbb{P}\{ζ>t\}$，其中$λ$是一个杀伤率参数，和$ψ:\mathbb{R}\to[0,1]$是一个非增函数。证明了若$ψ$是适当光滑的，函数$t\mapsto \mathbb{P}\{ζ>t\}$是二次连续可微的，且条件$0<-\frac{d\log\mathbb{P}\{ζ>t\}}{dt}<λ$对$ζ$的风险率成立，则存在唯一的连续可微函数$b$解光滑问题.我们展示了如何这一结果导致灵活的违约模型，它是可能的或有债权的期望值计算。
英文摘要：
The inverse first passage time problem asks whether, for a Brownian motion $B$ and a nonnegative random variable $ζ$, there exists a time-varying barrier $b$ such that $\mathbb{P}\{B_s>b(s),0\leq s\leq t\}=\mathbb{P}\{ζ>t\}$. We study a "smoothed" version of this problem and ask whether there is a "barrier" $b$ such that $ \mathbb{E}[\exp(-λ\int_0^tψ(B_s-b(s))\,ds)]=\mathbb{P}\{ζ>t\}$, where $λ$ is a killing rate parameter, and $ψ:\mathbb{R}\to[0,1]$ is a nonincreasing function. We prove that if $ψ$ is suitably smooth, the function $t\mapsto \mathbb{P}\{ζ>t\}$ is twice continuously differentiable, and the condition $0<-\frac{d\log\mathbb{P}\{ζ>t\}}{dt}<λ$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
