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

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Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK

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School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK

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Southampton Business School, University of Southampton, Southampton SO17 1BJ, UK

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Author to whom correspondence should be addressed.

Received: 27 May 2019 / Revised: 18 July 2019 / Accepted: 30 July 2019 / Published: 5 August 2019

(This article belongs to the Special Issue Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics)

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.