We study the $L^1$-approximation of the log-Heston SDE at the terminal time point by arbitrary methods that use an equidistant discretization of the driving Brownian motion. We show that such methods can achieve at most order $ \min \{ \nu, \tfrac{1}{2} \}$, where $\nu$ is the Feller index of the underlying CIR process. As a consequence Euler-type schemes are optimal for $\nu \geq 1$, since they have convergence order $\tfrac{1}{2}-\epsilon$ for $\epsilon >0$ arbitrarily small in this regime.

翻译：我们研究了在终端时间点，通过任意使用驱动布朗运动等距离散化的方法对log-Heston SDE进行$L^1$逼近的问题。我们证明了此类方法至多能达到阶$\min \{ \nu, \tfrac{1}{2} \}$，其中$\nu$是底层CIR过程的Feller指数。因此，对于$\nu \geq 1$的情况，欧拉型方法是最优的，因为在此条件下，对于任意小的$\epsilon >0$，其收敛阶为$\tfrac{1}{2}-\epsilon$。