Explicit solution of an inverse first-passage time problem for L\'{e}vy processes and counterparty credit risk

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 $\tau_b=\inf\{t\ge0: X(t) < b(t)\}$ 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 $\mu$ 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 L\'{e}vy process $X$ satisfying an exponential moment condition, we derive the solution of this problem in terms of $\lambda$-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.