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

Abstract

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.

18 pages