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Entry and exit decision problem with implementation delay. (English) Zbl 1167.60008
The paper deals with a problem of investment and disinvestment decisions [see K.A. Brekke and B. Øksendal, SIAM J. Control Optimization 32, No. 4, 1021–1036 (1994; Zbl 0801.60036)] for a rigorous mathematical treatment of the problem] in situations where there is a time lag \(d > 0\) from the time \(t\) when the decision is taken to the time \(t + d\) when the decision is implemented. The probabilistic approach to the combined entry and exit decisions under the Parisian implementation delay is applied. Among others it is proved the independence between Parisian stopping times which were introduced by M. Chesney, M. Jeanblanc-Picqué and M. Yor [Adv. Appl. Probab. 29, No. 1, 165–184 (1997; Zbl 0882.60042)] and a general Brownian motion with drift stopped at the stopping time. It is also solved the constrained maximization problem which allows to obtain an analytic solution to the optimal ‘starting’ and ‘stopping’ levels. The results have been compared with the instantaneous entry and exit situation, and show that an increase in the uncertainty of the underlying process hastens the decision to invest or disinvest. It extends a result of A. Bar-Ilan and W.J. Strange [Urban Econ. 39, No. 1, 87–113 (1996; Zbl 0915.90068)].

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
60J65 Brownian motion
62L15 Optimal stopping in statistics
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