Boundary classification and simulation of one-dimensional diffusion processes

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.