Projection scheme for polynomial diffusions on the unit ball

In this article, we consider numerical schemes for polynomial diffusions on the unit ball $\mathscr{B}^{d}$, which are solutions of stochastic differential equations with a diffusion coefficient of the form $\sqrt{1-|x|^{2}}$. We introduce a projection scheme on the unit ball $\mathscr{B}^{d}$ based on a backward Euler--Maruyama scheme and provide the $L^{2}$-rate of convergence. The main idea to consider the numerical scheme is the transformation argument introduced by Swart [29] for proving the pathwise uniqueness for some stochastic differential equation with a non-Lipschitz diffusion coefficient.