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On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes

1
Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK
2
School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK
3
Southampton Business School, University of Southampton, Southampton SO17 1BJ, UK
*
Author to whom correspondence should be addressed.
Risks 2019, 7(3), 87; https://doi.org/10.3390/risks7030087
Received: 27 May 2019 / Revised: 18 July 2019 / Accepted: 30 July 2019 / Published: 5 August 2019
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. View Full-Text
Keywords: Laplace transform; first hitting time; diffusion-type process; running maximum and minimum processes; boundary-value problem; normal reflection Laplace transform; first hitting time; diffusion-type process; running maximum and minimum processes; boundary-value problem; normal reflection
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Gapeev, P.V.; Rodosthenous, N.; Chinthalapati, V.L.R. On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes. Risks 2019, 7, 87.

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