On the Euler--Maruyama scheme for degenerate stochastic differential equations with non-sticky condition

The aim of this paper is to study weak and strong convergence of the Euler--Maruyama scheme for a solution of one-dimensional degenerate stochastic differential equation $mathrm{d} X_t=sigma(X_t) mathrm{d} W_t$ with non-sticky condition. For proving this, we first prove that the Euler--Maruyama scheme also satisfies non-sticky condition. As an example, we consider stochastic differential equation $mathrm{d} X_t=|X_t|^{alpha} mathrm{d} W_t$, $alpha in (0,1/2)$ with non-sticky boundary condition and we give some remarks on CEV models in mathematical finance.