Lévy过程和交易对手信用风险的首次穿越时间反问题的显式解
Explicit solution of an inverse first-passage time problem for Lévy processes and counterparty credit risk
摘要：
对于给定的马尔可夫过程$X$和$\mathbb{R}^+$上的生存函数$\overline{H}$，首次穿越时间逆问题（IFPT）是寻找一个障碍函数$b:\mathbb{R}^+\to[-\infty,+\infty]$，使得首次穿越时间$τ_b=\inf \{t\ge0:X(t)<b(t)\}$的生存函数由$\overline{H}$给出.在本文中，我们考虑一个版本的IFPT问题的障碍是固定为零，问题是要找到一个初始分布$μ$和时变$I$，使时变过程$X\circ I$的IFPT问题是解决了一个恒定的障碍在水平零。对于任何满足指数矩条件的Lévy过程$X$，我们根据过程$X$在第一次进入负半轴时的$λ$-不变分布导出了该问题的解.我们提供了这样的分布，这是一个独立的利益的结果明确的特征。对于给定的广义脆弱型多变量生存函数$\overline{H}$，我们随后构造了相应的IFPT的显式解，其中障碍水平固定为零。我们将该等结果应用于受交易对手信贷风险影响的金融合约的估值。
英文摘要：
For a given Markov process $X$ and survival function $\overline{H}$ on $\mathbb{R}^+$, the inverse first-passage time problem (IFPT) is to find a barrier function $b:\mathbb{R}^+\to[-\infty,+\infty]$ such that the survival function of the first-passage time $τ_b=\inf \{t\ge0:X(t)<b(t)\}$ is given by $\overline{H}$. In this paper, we consider a version of the IFPT problem where the barrier is fixed at zero and the problem is to find an initial distribution $μ$ and a time-change $I$ such that for the time-changed process $X\circ I$ the IFPT problem is solved by a constant barrier at the level zero. For any Lévy process $X$ satisfying an exponential moment condition, we derive the solution of this problem in terms of $λ$-invariant distributions of the process $X$ killed at the epoch of first entrance into the negative half-axis. We provide an explicit characterization of such distributions, which is a result of independent interest. For a given multi-variate survival function $\overline{H}$ of generalized frailty type, we construct subsequently an explicit solution to the corresponding IFPT with the barrier level fixed at zero. We apply these results to the valuation of financial contracts that are subject to counterparty credit risk.
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