Volume 23, Issue 1

In this paper, we study a third weak order scheme for diffusion processes which has been introduced by Alfonsi [1]. This scheme is built using cubature methods and is well defined under an abstract commutativity condition on the coefficients of the underlying diffusion process. Moreover, it has been proved in [1] that the third weak order convergence takes place for smooth test functions. First, we provide a necessary and sufficient explicit condition for the scheme to be well defined when we consider the one-dimensional case. In a second step, we use a result from [3] and prove that, under an ellipticity condition, this convergence also takes place for the total variation distance with order 3. We also give an estimate of the density function of the diffusion process and its derivatives.

Methods to estimate surface areas of geometric objects in 3D are well known. A number of these methods are of Monte Carlo type, and some are based on the Cauchy–Crofton formula from integral geometry. Employing this formula requires the generation of sets of random lines that are uniformly distributed in 3D. One model to generate sets of random lines that are uniformly distributed in 3D is called the tangent model (see [4]). In this paper, we present an extension of this model to higher dimensions, and we examine its performance by estimating hypersurface areas of n -ellipsoids. Then we apply this method to estimate surface areas of hypersurfaces defined by Fermat-type varieties of even degree.

We design and analyze an algorithm for estimating the mean of a function of a conditional expectation when the outer expectation is related to a rare event. The outer expectation is evaluated through the average along the path of an ergodic Markov chain generated by a Markov chain Monte Carlo sampler. The inner conditional expectation is computed as a non-parametric regression, using a least-squares method with a general function basis and a design given by the sampled Markov chain. We establish non-asymptotic bounds for the L2${L_{2}}$-empirical risks associated to this least-squares regression; this generalizes the error bounds usually obtained in the case of i.i.d. observations. Global error bounds are also derived for the nested expectation problem. Numerical results in the context of financial risk computations illustrate the performance of the algorithms.

We aim at analyzing in terms of a.s. convergence and weak rate the performances of the Multilevel Monte Carlo estimator (MLMC) introduced in [7] and of its weighted version, the Multilevel Richardson–Romberg estimator (ML2R), introduced in [12]. These two estimators permit to compute a very accurate approximation of I0=𝔼⁢[Y0]${I_{0}=\mathbb{E}[Y_{0}]}$ by a Monte Carlo-type estimator when the (non-degenerate) random variable Y0∈L2⁢(ℙ)${Y_{0}\in L^{2}(\mathbb{P})}$ cannot be simulated (exactly) at a reasonable computational cost whereas a family of simulatable approximations (Yh)h∈ℋ${(Y_{h})_{h\in\operatorname{\mathcal{H}}}}$ is available. We will carry out these investigations in an abstract framework before applying our results, mainly a Strong Law of Large Numbers and a Central Limit Theorem, to some typical fields of applications: discretization schemes of diffusions and nested Monte Carlo.