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On perpetual American put valuation and first-passage in a regime-switching model with jumps. (English) Zbl 1164.60066
The authors introduce a continuous-time irreducible Markov process $$Z$$ with finite state space and a regime-switching phase-type Lévy process $$X$$, controlled by $$Z$$. They consider financial markets with bond and stock, where stock is an exponent of $$X$$. This consideration is motivated by the observation that Lévy processes have been successfully calibrated to options with single, short time maturities whereas regime-switching models fit well longer dated options. Under this model, for the case of the dense class of phase-type jumps, explicit and analytically tractable results are obtained for the value function of a perpetual American put option and for the corresponding optimal exercise strategy. It is proved that the optimal stopping time takes the form of the first-passage problem of $$X$$ under some level $$k(Z_t)$$, and the corresponding first-passage problem is solved. The solution is supported by path transformation and a new matrix Wiener-Hopf factorization result for this class of processes.

##### MSC:
 60K15 Markov renewal processes, semi-Markov processes 91B28 Finance etc. (MSC2000)
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