The author's presentation of multilevel Monte Carlo path simulation at the MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo methods. This paper reviews the progress since then, emphasising the simplicity, flexibility and generality of the multilevel Monte Carlo approach. It also offers a few original ideas and suggests areas for future research.
The Monte Carlo complexity of computing integrals depending on a parameter is analyzed for smooth integrands. An optimal algorithm is developed on the basis of a multigrid variance reduction technique. The complexity analysis implies that our algorithm attains a higher convergence rate than any deterministic algorithm. Moreover, because of savings due to computation on multiple grids, this rate is also higher than that of previously developed Monte Carlo algorithms for parametric integration.  The problem of the global solution of Fredholm integral equations is studied. This means that one seeks to approximate the full solution function (as opposed to the local problem, where only the value of the solution in a single point or a functional of the solution is sought). The Monte Carlo complexity, i.e., the complexity of the stochastic solution of this problem, is analyzed. The framework for this analysis is provided by information-based complexity theory. The investigations complement previous ones on the stochastic complexity of the local solution and on deterministic complexity of both local and global solutions. The results show that even in the global case Monte Carlo algorithms can perform better than deterministic ones, although the difference is not as large as in the local case. We study the randomized approximation of weakly singular integral operators. For a suitable class of kernels having a standard type of singularity and being otherwise of finite smoothness, we develop a Monte Carlo multilevel method, give convergence estimates and prove lower bounds which show the optimality of this method and establish the complexity. As an application we obtain optimal methods for and the complexity of randomized solution of the Poisson equation in simple domains, when the solution is sought on subdomains of arbitrary dimension. Approximation properties of the underlying estimator are used to improve the efficiency of the method of dependent tests. A multilevel approximation procedure is developed such that in each level the number of samples is balanced with the level-dependent variance, resulting in a considerable reduction of the overall computational cost. The new technique is applied to the Monte Carlo estimation of integrals depending on a parameter.  In this paper we show that the Milstein scheme can be used to improve the convergence of the multilevel Monte Carlo method for scalar stochastic differential equations. Numerical results for Asian, lookback, barrier and digital options demonstrate that the computational cost to achieve a root-mean-square error of e is reduced to O(e -2). This is achieved through a careful construction of the multilevel estimator which computes the difference in expected payoff when using different numbers of timesteps.