2013/08/02 by Masahiko Egami, Egami, Masahiko, Tadao Oryu +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Primary: 60G40 #Probability (math.PR) #Probability and Risk Models #Secondary: 60J75 #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60G40 #msc:60J75
paper · pdf · doi:10.48550/arxiv.1308.0509
21 pages, 11 figures
openalex publication_date 2013/08/02 · arxiv created 2015/04/14 · arxiv updated 2015/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a new type of optimal stopping problems where the absorbing boundary moves as the state process X attains new maxima S. More specifically, we set the absorbing boundary as S-b where b is a certain constant. This problem is naturally connected with excursions from zero of the reflected process S-X. We examine this constrained optimization with the state variable X as a spectrally negative Levy process. The problem is in nature a two-dimensional one. The threshold strategy given by the path of X is not in fact optimal. It turns out, however, that we can reduce the original problem to an infinite number of one-dimensional optimal stopping problems, and we find explicit solutions. This work is motivated by the bank's profit maximization with the constraint that it maintain a certain level of leverage ratio. When the bank's asset value severely deteriorates, the bank's required capital requirement shall be violated. This situation corresponds to X<S-b in our setting. This model may well describe a real-life situation where even a big bank can fail because the absorbing boundary is keeping up with the size of the bank.