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On the laplace transforms of the first hitting times for drawdowns and drawups of diffusion-type processes

On the laplace transforms of the first hitting times for drawdowns and drawups of diffusion-type processes
On the laplace transforms of the first hitting times for drawdowns and drawups of diffusion-type processes

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.

Boundary-value problem, Diffusion-type process, First hitting time, Laplace transform, Normal reflection, Running maximum and minimum processes
2227-9091
1-15
Gapeev, Pavel V.
2d5a93ce-753b-449b-9533-cddba5c3a179
Rodosthenous, Neofytos
97ab9e6b-ff80-4e7d-866b-43708fb6bcce
Raju Chinthalapati, V.L.
65ec749f-9695-4550-a408-93c649f807af
Gapeev, Pavel V.
2d5a93ce-753b-449b-9533-cddba5c3a179
Rodosthenous, Neofytos
97ab9e6b-ff80-4e7d-866b-43708fb6bcce
Raju Chinthalapati, V.L.
65ec749f-9695-4550-a408-93c649f807af

Gapeev, Pavel V., Rodosthenous, Neofytos and Raju Chinthalapati, V.L. (2019) On the laplace transforms of the first hitting times for drawdowns and drawups of diffusion-type processes. Risks, 7 (3), 1-15, [87]. (doi:10.3390/risks7030087).

Record type: Article

Abstract

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.

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More information

Accepted/In Press date: 30 July 2019
e-pub ahead of print date: 5 August 2019
Published date: 1 September 2019
Keywords: Boundary-value problem, Diffusion-type process, First hitting time, Laplace transform, Normal reflection, Running maximum and minimum processes

Identifiers

Local EPrints ID: 433704
URI: http://eprints.soton.ac.uk/id/eprint/433704
ISSN: 2227-9091
PURE UUID: ea1afd0a-1ff0-4de5-8f34-11920ce79130

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Date deposited: 02 Sep 2019 16:30
Last modified: 27 Apr 2022 07:16

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Contributors

Author: Pavel V. Gapeev
Author: Neofytos Rodosthenous

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