EXPLICIT SOLUTION OF AN INVERSE FIRST-PASSAGE TIME PROBLEM FOR LÉVY PROCESSES AND COUNTERPARTY CREDIT RISK

• Published in: The Annals of applied probability 2015-10, Vol.25 (5), p.2383-2415
• Main Authors: Davis, M. H. A., Pistorius, M. R.
• Format: Article
• Language: English
• Subjects: 60J75; 91G40; 91G80; counterparty risk; Credit risk; Inverse first-passage problem; Lévy process; Markov analysis; Mathematical functions; Mathematical problems; multi-variate first-passage times; quasi-invariant distribution
• ISSN: 1050-5164; 2168-8737
• DOI: 10.1214/14-AAP1051

For a given Markov process X and survival function H̄ on ℝ+, the inverse first-passage time problem (IFPT) is to find a barrier function b:ℝ+ → [−∞, +∞] such that the survival function of the first-passage time τb = inf{t ≥ 0:X(t) < b(t)} is given by H̄. 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 ο 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 H̄ 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.