Article: Boundary classification and simulation of one-dimensional diffusion processes Journal: International Journal of Mathematics in Operational Research (IJMOR) 2017 Vol.11 No.1 pp.107 - 138 Abstract: Diffusion processes are of great interest both in theoretical and applied mathematics. They may approximate or modelise many physical, biological, economic, and social phenomena. Their behaviour on or near the endpoints of any given initialisation interval of the state space, is an important notion which plays a fundamental role in mathematical modelling. In order to deal with the comportment of diffusion processes at boundary points, we follow an analytic delineation which uses an appropriate second-order differential operator (the basic infinitesimal operator of the process) coupled with boundary conditions. The nature of these boundary constraints delimits the boundary classification of the diffusion process. We make in this paper an overview on the modern classifications of possible behaviour near the boundaries of a given sub-interval of the state space. We also shed light on simulation of one-dimensional diffusion processes using discretisation techniques. Inderscience Publishers - linking academia, business and industry through research

Title: Boundary classification and simulation of one-dimensional diffusion processes

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 Monasti, 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, Unité de Recherche 'Multifractales Et Ondelettes' (UR11ES53), Faculté des Sciences de Monastir, Université de Monastir, Avenue de l'Environnement, Monastir, 5000, Tunisia

Abstract: Diffusion processes are of great interest both in theoretical and applied mathematics. They may approximate or modelise many physical, biological, economic, and social phenomena. Their behaviour on or near the endpoints of any given initialisation interval of the state space, is an important notion which plays a fundamental role in mathematical modelling. In order to deal with the comportment of diffusion processes at boundary points, we follow an analytic delineation which uses an appropriate second-order differential operator (the basic infinitesimal operator of the process) coupled with boundary conditions. The nature of these boundary constraints delimits the boundary classification of the diffusion process. We make in this paper an overview on the modern classifications of possible behaviour near the boundaries of a given sub-interval of the state space. We also shed light on simulation of one-dimensional diffusion processes using discretisation techniques.

Keywords: linear diffusion process; natural; exit; entrance; attainable; unattainable; attracting and regular boundary; hitting time; scale function; boundary classification; weak solution; strong solution; simulation; geometric Brownian motion; Euler-Maruyama algorithm; Milstein algorithm.

DOI: 10.1504/IJMOR.2017.085382

International Journal of Mathematics in Operational Research, 2017 Vol.11 No.1, pp.107 - 138

Received: 05 Aug 2015
Accepted: 07 Nov 2015
Published online: 25 Jul 2017 *

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Title: Boundary classification and simulation of one-dimensional diffusion processes

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