We analyse a multilevel Monte Carlo method for the approximation of distribution functions of univariate random variables. Since, by assumption, the target distribution is not known explicitly, approximations have to be used. We provide an asymptotic analysis of the error and the cost of the algorithm. Furthermore we construct an adaptive version of the algorithm that does not require any a priori knowledge on weak or strong convergence rates. We apply the adaptive algorithm to smooth path-independent and path-dependent functionals and to stopped exit times of SDEs.
As the size of engineered systems grows, problems in reliability theory can become computationally challenging, often due to the combinatorial growth in the number of cut sets. In this paper we demonstrate how Multilevel Monte Carlo (MLMC) -a simulation approach which is typically used for stochastic differential equation models -can be applied in reliability problems by carefully controlling the bias-variance tradeoff in approximating large system behaviour. In this first exposition of MLMC methods in reliability problems we address the canonical problem of estimating the expectation of a functional of system lifetime for non-repairable and repairable components, demonstrating the computational advantages compared to classical Monte Carlo methods. The difference in computational complexity can be orders of magnitude for very large or complicated system structures, or where the desired precision is lower.
In this paper we present a novel approach towards variance reduction for
discretised diffusion processes. The proposed approach involves specially
constructed control variates and allows for a significant reduction in the
variance for the terminal functionals. In this way the complexity order of the
standard Monte Carlo algorithm ($\varepsilon^{-3}$ in the case of a first order
scheme and $\varepsilon^{-2.5}$ in the case of a second order scheme) can be
reduced down to $\varepsilon^{-2+\delta}$ for any $\delta\in [0,0.25)$ with
$\varepsilon$ being the precision to be achieved. These theoretical results are
illustrated by several numerical examples
In this article we propose a novel approach to reducing the computational complexity of various approximation methods for pricing discrete time American or Bermudan options. Given a sequence of continuation values estimates corresponding to different levels of spatial approximation, we propose a multilevel low biased estimate for the price of the option. It turns out that the resulting complexity gain can be of order ε −1 with ε denoting the desired precision. The performance of the proposed multilevel algorithms is illustrated by a numerical example.
In this paper we discuss the possibility of using multilevel Monte Carlo (MLMC) methods for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. We exemplify this general idea in the case of weak Euler scheme for Lévy driven stochastic differential equations, and show that, given a weak convergence of order α ≥ 1/2, the complexity of the corresponding "weak" MLMC estimate is of order −2 log 2 ( ). The numerical performance of the new "weak" MLMC method is illustrated by several numerical examples.
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