We consider the problem of the discrete-time approximation of the solution of a one-dimensional SDE with piecewise locally Lipschitz drift and continuous diffusion coefficients with polynomial growth. In this paper, we study the strong convergence of a (semi-explicit) exponential-Euler scheme previously introduced in Bossy et al. (2021). We show the usual 1/2 rate of convergence for the exponential-Euler scheme when the drift is continuous. When the drift is discontinuous, the convergence rate is penalised by a factor {$\epsilon$} decreasing with the time-step. We examine the case of the diffusion coefficient vanishing at zero, which adds a positivity preservation condition and a convergence analysis that exploits the negative moments and exponential moments of the scheme with the help of change of time technique introduced in Berkaoui et al. (2008). Asymptotic behaviour and theoretical stability of the exponential scheme, as well as numerical experiments, are also presented.

翻译：我们考虑一类具有分段局部Lipschitz连续漂移和多项式增长连续扩散系数的一维随机微分方程解的离散时间逼近问题。本文研究Bossy等人(2021)提出的(半显式)指数Euler格式的强收敛性。当漂移连续时，我们证明该格式具有通常的1/2收敛阶。当漂移不连续时，收敛速度受到一个随步长减小而递减的因子{$\epsilon$}的惩罚。我们考察扩散系数在零点消逝的情形，此时需附加正性保持条件，并借助Berkaoui等人(2008)引入的时间变换技术，利用格式的负矩与指数矩进行收敛性分析。本文同时给出了指数格式的渐近行为与理论稳定性分析以及数值实验。