Authors: Noureddine Jilani Ben Naouara; Faouzi Trabelsi
Addresses: Department of Mathematics, Unité de Recherche 'Multifractales Et Ondelettes' (UR11ES53), Faculté des Sciences de Monastir, Université de Monastir, Avenue de l'Environnement, Monastir 5000, Tunisia ' Department of Mathematics, Institut Supérieur d'Informatique et de Mathématiques de Monastir, Avenue de la Korniche, B.P. 223, 5000 Monastir, Tunisia; Unité de Recherche 'Multifractales Et Ondelettes' (UR11ES53), Faculté des Sciences de Monastir, Université de Monastir, Avenue de l'Environnement, Monastir 5000, Tunisia
Abstract: In this paper, we develop some extensions of results proposed by Dayanik and Karatzas (2003) on optimal stopping time problem, and apply them to the Wright-Fisher diffusion process. In this respect, we study in details the important problem of boundary behaviour of the diffusion that approximates the Markov chain. As application, we find the perfect time to turn off the monitor genetic diseases. Precisely, we find the optimal stopping time at which is maximised the probability that a sample of size n, taken from large infected population, consists only of uninfected individuals. We show that this maximal probability is nothing other than the probability of fixation of the infected allele in case of bounded scale function.
Keywords: optimal stopping time; linear diffusion; natural boundary; exit boundary; entrance boundary; regular boundary; Wright-Fisher diffusion process; haploid; diploid; DNA sequencing; genetic diseases.
International Journal of Mathematical Modelling and Numerical Optimisation, 2014 Vol.5 No.3, pp.229 - 264
Received: 26 Dec 2013
Accepted: 02 May 2014
Published online: 08 Jul 2014 *