We study discrete-time simulation schemes for stochastic Volterra equations, namely the Euler and Milstein schemes, and the corresponding Multilevel Monte-Carlo method. By using and adapting some results from Zhang (2008), together with the Garsia–Rodemich–Rumsey lemma, we obtain the convergence rates of the Euler scheme and Milstein scheme under the supremum norm. We then apply these schemes to approximate the expectation of functionals of such Volterra equations by the (Multilevel) Monte-Carlo method, and compute their complexity. We finally provide some numerical simulation results.

中文翻译：

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随机 Volterra 方程的离散时间模拟

我们研究随机 Volterra 方程的离散时间模拟方案，即 Euler 和 Milstein 方案，以及相应的多级蒙特卡洛方法。通过使用和改编 Zhang (2008) 的一些结果，结合 Garsia-Rodemich-Rumsey 引理，我们得到了欧拉方案和米尔斯坦方案在最高范数下的收敛速度。然后我们应用这些方案通过（多级）蒙特卡洛方法来近似这些沃尔泰拉方程的泛函期望，并计算它们的复杂性。我们最后提供了一些数值模拟结果。