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Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. (English) Zbl 1042.60023
The authors consider spectrally negative Levý processes (these are real-valued random processes with stationary independent increments which have no positive jumps) and determine the joint Laplace transform of the exit time and the exit position from the interval containing the origin of the process reflected in its supremum. This Laplace transform can be written in terms of scale functions that already appeared as a solution to the two-sided exit problem. An optimal stopping problem is outlined which is associated with the price of Russian options. This optimal stopping problem is solved in terms of scale functions which appear in the afore mentioned exit problems. A modification of the optimal stopping problem known as Canadization is considered and it is shown that an explicit solution is also available in terms of scale functions. The paper is concluded with some examples of optimal stopping problem.

60G51 Processes with independent increments; Lévy processes
60J99 Markov processes
60G40 Stopping times; optimal stopping problems; gambling theory
91B70 Stochastic models in economics
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