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Optimal stopping problems for the maximum process with upper and lower caps. (English) Zbl 1290.60048

Let \(X\) be a spectrally negative Lévy process, and \(\bar{X}_t=s\vee\sup_{0\leq u\leq t}X_u\). The author is interested in the following optimal stopping problem: \(V_\varepsilon^*(x,s)=\sup_{\tau\in \mathcal{M}}\operatorname{E}_{x,s}[e^{-q\tau +\bar{X}_\tau\wedge\varepsilon}]\) for \(t\geq 0\), \(s\geq x\). Since the constant \(\varepsilon\) bounds the process \(\bar{X}\) from above, it is referred to as the upper cup. So, a capped version of the Shepp-Shiryaev optimal stopping problem is considered. The method of solving the optimal stopping problem consists of a verification technique, that is, a candidate solution is heuristically derived and then it is verified that it is indeed a solution. The solution is provided explicitly in terms of scale functions. The optimal stopping boundary is characterized by an ordinary differential equation involving scale functions. A modification with the lower cap is also considered. Whilst this is already included in the starting point of the maximum process \(\bar{X}\), there is a stopping problem that captures this idea of lower cap in the sense that the decision to exercise has to be made before \(X\) drops below a certain level. After proving the main results, some examples are considered, including Brownian motion with drift and with compound Poisson jumps and also stable processes.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60G51 Processes with independent increments; Lévy processes
60J75 Jump processes (MSC2010)

Software:

mlf; Mittag-Leffler
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Full Text: DOI arXiv

References:

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