Volume 24, Issue 4

A new method based on nesting Monte Carlo is developed to solve high-dimensional semi-linear PDEs. Depending on the type of non-linearity, different schemes are proposed and theoretically studied: variance error are given and it is shown that the bias of the schemes can be controlled. The limitation of the method is that the maturity or the Lipschitz constants of the non-linearity should not be too high in order to avoid an explosion of the computational time. Many numerical results are given in high dimension for cases where analytical solutions are available or where some solutions can be computed by deep-learning methods.

We present the Euler–Maruyama approximation for one-dimensional stochastic differential equations involving the local time at point zero. Also, we prove the strong convergence of the Euler–Maruyama approximation whose both drift and diffusion coefficients are Lipschitz. After that, we generalize to the non-Lipschitz case.

The paper is devoted to the modeling of a single-phase flow through saturated porous media. A statistical approach where permeability is considered as a lognormal random field is applied. The impact of permeability, random boundary conditions and wells pressure on the flow in a production well is studied. A numerical procedure to generate an ensemble of realizations of the numerical solution of the problem is developed. A global sensitivity analysis is performed using Sobol indices. The impact of different model parameters on the total model uncertainty is studied.

Synthetic Aperture Radar (SAR) images are widely used in several environmental applications because they provide information which cannot be obtained with other sensors. The 𝒢I0{\mathcal{G}_{I}^{0}} distribution is an important model for these images because of its flexibility (it provides a suitable way for modeling areas with different degrees of texture, reflectivity and signal-to-noise ratio) and tractability (it is closely related to the Snedekor-F, Pareto Type II, and Gamma distributions). Simulated data are important for devising tools for SAR image processing, analysis and interpretation, among other applications. We compare four ways for sampling data that follow the 𝒢I0{\mathcal{G}_{I}^{0}} distribution, using several criteria for assessing the quality of the generated data and the consumed processing time. The experiments are performed running codes in four different programming languages. The experimental results indicate that although there is no overall best method in all the considered programming languages, it is possible to make specific recommendations for each one.

This paper proposes a new Markov chain approach to second-order weak approximations of stochastic differential equations (SDEs) driven by d -dimensional Brownian motion. The scheme is explicitly constructed by polynomials of Brownian motions up to second order, and any discrete moment-matched random variables or the Lévy area simulation method are not used. The required number of random variables is still d in one-step simulation of the implementation of the scheme. In the Markov chain, a correction term with Lie bracket of vector fields associated with SDEs appears as the cost of not using moment-matched random variables.

We explore different methods of solving systems of stochastic differential equations by first implementing the Euler–Maruyama and Milstein methods with a Monte Carlo simulation on a CPU. The performance of the methods is significantly improved through the recently developed antithetic multilevel Monte Carlo estimator, which yields a computation complexity of 𝒪⁢(ϵ-2){\mathcal{O}(\epsilon^{-2})} root-mean-square error and does so without the approximation of Lévy areas. Further improvements in performance are gained by moving the algorithms to a GPU - first on a single device and then on a multi-GPU cluster. Our GPU implementation of the antithetic multilevel Monte Carlo displays a major speedup in computation when compared with many commonly used approaches in the literature. While our work is focused on the simulation of the stochastic volatility and interest rate model, it is easily extendable to other stochastic systems, and it is of particular interest to those with non-diagonal, non-commutative noise.

The influence of boundary singularities on the behavior of the random walk algorithms is studied and illustrated by the results of several computational experiments of solving boundary-value problems for elliptic equations.