Volume 20, Issue 3

Certain very high-dimensional integrals can be efficiently approximated by quasi-Monte Carlo (q-MC) methods. If the average dimension of the integrand is small, the convergence rate can be near to 1/ N , where N is the sample size.

We present in this paper a further development of the stochastic spectral method for solving boundary value problems in domains which are composed by a set of overlapped discs first suggested by the first author in Appl. Math. Comput. 219 (2013), no. 10, 5123–5139]. We study statistical characteristics of the solution to isotropic diffusion problem in response to fluctuating incoming flux of particles through the circular-shaped boundaries. Performance of the method is illustrated by a series of numerical experiments. The method can be considered as a direct inversion of the integral Poisson formula representing the solution in the disc, so it is highly accurate and fast for the class of domains considered. This makes possible to solve an ensemble of equations with random samples of boundary conditions and calculate the desired statistical characteristics.

The purpose of this paper is to study the problem of pricing Asian options using the multilevel Monte Carlo method recently introduced by Giles [Oper. Res. 56 (2008), no. 3, 607–617] and to prove a central limit theorem of Lindeberg–Feller type for the obtained algorithm. Indeed, the implementation of such a method requires first a discretization of the integral of the payoff process. For this, we use two well-known second order discretization schemes, namely, the Riemann scheme and the trapezoidal scheme. More precisely, for each of these schemes, we prove a stable law convergence result for the error on two consecutive levels of the algorithm. This allows us to go further and prove two central limit theorems on the multilevel algorithm providing us a precise description on the choice of the associated parameters with an explicit representation of the limiting variance. For this setting of second order schemes, we give new optimal parameters leading to the convergence of the central limit theorem. The complexity of the multilevel Monte Carlo algorithm will be determined.

This paper is devoted to the simulation of the Credit Valuation Adjustment (CVA) using a pure Monte Carlo technique with Malliavin calculus (MCM). The procedure presented is based on a general theoretical framework that includes a large number of models as well as various contracts, and allows both the computation of CVA and its sensitivity with respect to the different assets. Moreover, we provide the expression of the backward conditional density of assets vector that can be simulated off-line in order to reduce the variance of the CVA estimator. Using the suitability of MCM to parallel architectures and thus to a Graphic Processing Unit (GPU) implementation, we show that the results obtained are accurate once a sufficient number of trajectories is simulated. Both complexity and accuracy are studied for MCM and regression methods and are compared to the square Monte Carlo benchmark.

In this paper, we directly obtain from Monte Carlo simulations the critical magnetic field H=0.29(3)${H=0.29(3)}$ of the field-induced Kosterlitz–Thouless transition in the critical triangular-lattice antiferromagnet. The Yang–Lee zero approach clearly shows the field-induced Kosterlitz–Thouless transition and the critical magnetic field agrees well with the results from other indirect methods. Also, the reduced zero-field susceptibility gives us the same conclusion. For the investigations, we used the exact and approximate ground densities of states as a function of magnetization by using both the exact enumeration method for small systems (up to 9×9 lattices) and the Wang–Landau Monte Carlo algorithm for large systems (up to 30×30 lattices).