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Erscheinungsjahr: 
2019
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[Journal:] Risks [ISSN:] 2227-9091 [Volume:] 7 [Issue:] 3 [Article No.:] 87 [Publisher:] MDPI [Place:] Basel [Year:] 2019 [Pages:] 1-15
Verlag: 
MDPI, Basel
Zusammenfassung: 
We obtain closed-form expressions for the value of the joint Laplace transform of the running maximum and minimum of a diffusion-type process stopped at the first time at which the associated drawdown or drawup process hits a constant level before an independent exponential random time. It is assumed that the coefficients of the diffusion-type process are regular functions of the current values of its running maximum and minimum. The proof is based on the solution to the equivalent inhomogeneous ordinary differential boundary-value problem and the application of the normal-reflection conditions for the value function at the edges of the state space of the resulting three-dimensional Markov process. The result is related to the computation of probability characteristics of the take-profit and stop-loss values of a market trader during a given time period.
Schlagwörter: 
boundary-value problem
diffusion-type process
first hitting time
Laplace transform
normal reflection
running maximum and minimum processes
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