## The Annals of Applied Probability

### Explicit solution of an inverse first-passage time problem for Lévy processes and counterparty credit risk

#### Abstract

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 Lévy 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.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 5 (2015), 2383-2415.

Dates
Revised: March 2014
First available in Project Euclid: 30 July 2015

https://projecteuclid.org/euclid.aoap/1438261044

Digital Object Identifier
doi:10.1214/14-AAP1051

Mathematical Reviews number (MathSciNet)
MR3375879

Zentralblatt MATH identifier
1325.60071

#### Citation

Davis, M. H. A.; Pistorius, M. R. Explicit solution of an inverse first-passage time problem for Lévy processes and counterparty credit risk. Ann. Appl. Probab. 25 (2015), no. 5, 2383--2415. doi:10.1214/14-AAP1051. https://projecteuclid.org/euclid.aoap/1438261044

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