On the construction of boundary preserving numerical schemes

Abstract

Our aim in this note is to extend the semi discrete technique by combine it with the split step method. We apply our new method to the Ait-Sahalia model and propose an explicit and positivity preserving numerical scheme.

Boundary preserving explicit scheme for the Aït-Sahalia mode

“…First, we use a set of parameters so that (35) works; we take the coefficients k −1 = 2, k 0 = 3, k 1 = 4, k 2 = 7, k 3 = 1 the exponents r = 2 and ρ = 3/2 and T = 1 (SET I). In other words we consider the critical case (3).…”

“…The model was proposed in [1] to study the spot interest rate and especially for the case r = 2. The numerical approximation of (1) was studied in [8] using an implicit scheme, in [6] again by applying an implicit method and in [3] using an explicit positivity preserving numerical scheme applying the semi-discrete method, see [2] for the original proposition of the method and [7] and references therein for a review of the method. The order of strong convergence of the proposed semi-discrete numerical scheme in [3] was not proved.…”

“…The numerical approximation of (1) was studied in [8] using an implicit scheme, in [6] again by applying an implicit method and in [3] using an explicit positivity preserving numerical scheme applying the semi-discrete method, see [2] for the original proposition of the method and [7] and references therein for a review of the method. The order of strong convergence of the proposed semi-discrete numerical scheme in [3] was not proved. Using a different approach here, that is approximating the Lamperti transformation of the original sde and then transforming back, we prove strong convergence of the method with order at least 1, see Section 3.1.…”

“…The cases studied in [8] and [3] correspond to coefficients satisfying 2ρ < r + 1. Here we study the critical case when 2ρ = r + 1 with r = 2, therefore the dynamics of x t are described by…”