This paper focuses on the study of an original combination of the Multilevel Monte Carlo method introduced by Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56(3) (2008), pp. 607–617.] and the popular importance sampling technique. To compute the optimal choice of the parameter involved in the importance sampling method, we rely on Robbins–Monro type stochastic algorithms. On the one hand, we extend our previous work [M. Ben Alaya, K. Hajji and A. Kebaier, Importance sampling and statistical Romberg method, Bernoulli 21(4) (2015), pp. 1947–1983.] to the Multilevel Monte Carlo setting. On the other hand, we improve [M. Ben Alaya, K. Hajji and A. Kebaier, Importance sampling and statistical Romberg method, Bernoulli 21(4) (2015), pp. 1947–1983.] by providing a new adaptive algorithm avoiding the discretization of any additional process. Furthermore, from a technical point of view, the use of the same stochastic algorithms as in [M. Ben Alaya, K. Hajji and A. Kebaier, Importance sampling and statistical Romberg method, Bernoulli 21(4) (2015), pp. 1947–1983.] appears to be problematic. To overcome this issue, we employ an alternative version of stochastic algorithms with projection (see, e.g. Laruelle, Lehalle and Pagès [Optimal posting price of limit orders: learning by trading, Math. Financ. Econ. 7(3) (2013), pp. 359–403.]). In this setting, we show innovative limit theorems for a doubly indexed stochastic algorithm which appear to be crucial to study the asymptotic behaviour of the new adaptive Multilevel Monte Carlo estimator. Finally, we illustrate the efficiency of our method through applications from quantitative finance.

本文重点研究了 Giles 引入的多级蒙特卡罗方法的原始组合 [多级蒙特卡罗路径模拟，Oper。水库。56(3) (2008), pp. 607–617.] 和流行的重要性采样技术。为了计算重要性采样方法中涉及的参数的最佳选择，我们依赖于 Robbins–Monro 类型的随机算法。一方面，我们扩展了我们之前的工作 [M. Ben Alaya、K. Hajji 和 A. Kebaier，重要性抽样和统计 Romberg 方法，Bernoulli 21(4) (2015)，pp. 1947–1983.] 到多级蒙特卡洛设置。另一方面，我们改进了 [M. Ben Alaya、K. Hajji 和 A. Kebaier，重要性抽样和统计 Romberg 方法, Bernoulli 21(4) (2015), pp. 1947–1983.] 通过提供一种新的自适应算法避免任何附加过程的离散化。此外，从技术角度来看，使用与 [M. Ben Alaya、K. Hajji 和 A. Kebaier，重要性抽样和统计 Romberg 方法，Bernoulli 21(4) (2015)，pp. 1947–1983.] 似乎有问题。为了克服这个问题，我们采用了带有投影的随机算法的替代版本（参见，例如 Laruelle、Lehalle 和 Pagès [限价订单的最佳发布价格：通过交易学习， 数学。财务。经济。7(3)（2013 年），第 359–403 页。]）。在这种情况下，我们展示了双索引随机算法的创新极限定理，这似乎对于研究新的自适应多级蒙特卡罗估计器的渐近行为至关重要。最后，我们通过量化金融的应用来说明我们方法的效率。