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

• 成果类型: Article
• 署名作者: Davis, M. H. A.; Pistorius, M. R.
• 署名单位: Imperial College London
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1051
• 发表日期: 2015
• 页码: 2383-2415
• 关键词: 1st passage

摘要

For a given Markov process X and survival function (H) over bar on R+, the inverse first-passage time problem (IFPT) is to find a bather function b : R+ -> [-infinity, +infinity] such that the survival function of the first-passage time tau(b) = inf{t >= 0: X (t) < b(t)} is given by <(H)over bar>. In this paper, we consider a version of the IFPT problem where the bather 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 circle I the IFPT problem is solved by a constant bather at the level zero. For any Levy 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 (H) over bar of generalized frailty type, we construct subsequently an explicit solution to the corresponding IFPT with the bather level fixed at zero. We apply these results to the valuation of financial contracts that are subject to counterparty credit risk.