optimal stopping when the absorbing boundary is following：吸收边界的最优停止.pdf

OPTIMAL STOPPING WHEN THE ABSORBING BOUNDARY IS FOLLOWING AFTER MASAHIKO EGAMI AND TADAO ORYU ABSTRACT. 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 just 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 re- quirement 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 large bank can fail because the absorbing boundary is keeping up with the size of the bank. ´ Key words: Optimal stopping; excursion theory; spectrally negative Levy processes; scale functions. Mathematics Subject Classification (2010) : Primary: 60G40 Secondary: 60J75 1. INTRODUCTION ´ We study a new type of optimal stopping