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

## TL;DR

This paper investigates the convergence properties of the Euler--Maruyama scheme applied to degenerate stochastic differential equations with non-sticky boundary conditions, including examples relevant to financial models.

## Contribution

It establishes that the Euler--Maruyama scheme preserves non-sticky conditions and analyzes its weak and strong convergence for degenerate SDEs with boundary constraints.

## Key findings

- Euler--Maruyama scheme satisfies non-sticky condition
- Convergence results for degenerate SDEs with non-sticky boundary
- Application to CEV models in finance

## 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.