Monte Carlo Finite Difference Method Research Articles

Multilevel Monte Carlo Finite Difference Methods for Fractional Conservation Laws with Random Data

We establish a notion of random entropy solution for degenerate fractional conservation laws incorporating randomness in the initial data, convective flux, and diffusive flux. In order to quantify the solution uncertainty, we design a multilevel Monte Carlo finite difference method (MLMC-FDM) to approximate the ensemble average of the random entropy solutions. Furthermore, we analyze the convergence rates for MLMC-FDM and compare them with the convergence rates for the deterministic case. Additionally, we formulate error vs. work estimates for the multilevel estimator. Finally, we present several numerical experiments to demonstrate the efficiency of these schemes and validate the theoretical estimates obtained in this work.

Nonparametric Malliavin–Monte Carlo Computation of Hedging Greeks

We propose a way to compute the hedging Delta using the Malliavin weight method. Our approach, which we name the λ-method, generally outperforms the standard Monte Carlo finite difference method, especially for discontinuous payoffs. Furthermore, our approach is nonparametric, as we only assume a general local volatility model and we substitute the volatility and the other processes involved in the Greek formula with quantities that can be nonparametrically estimated from a given time series of observed prices.

The POD–DEIM reduced-order method for stochastic Allen–Cahn equations with multiplicative noise

In this paper, we propose a reduced-order method (ROM) based on the Monte Carlo finite difference method (FDM) or Monte Carlo finite element method (FEM) for the stochastic Allen–Cahn (SAC) equation with multiplicative noise. The reduction method is a combination of proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). In the theoretical part, the error bounds of full-order method (FOM) and POD–DEIM are provided, as well as the computational complexity of FOM and POD–DEIM. In the numerical experiments section, several two-dimensional SAC examples with multiplicative noise are considered. Using the Monte Carlo FDM and Monte Carlo FEM as the reference methods, the POD–DEIM has obvious advantage in the computational efficiency. And with the encryption in the space or time direction, this advantage is more prominent.

A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations

This paper proposes a finite difference multilevel Monte Carlo algorithm for degenerate parabolic convection–diffusion equations where the convective and diffusive fluxes are allowed to be random. We establish a notion of stochastic entropy solutions to these equations. Our chief goal is to efficiently compute approximations to statistical moments of these stochastic entropy solutions. To this end, we design a multilevel Monte Carlo method based on a finite volume scheme for each sample. We present a novel convergence rate analysis of the combined multilevel Monte Carlo finite volume method, allowing in particular for low [Formula: see text]-integrability of the random solution with [Formula: see text], and low deterministic convergence rates (here, the theoretical rate is [Formula: see text]). We analyze the design and error versus work of the multilevel estimators. We obtain that the maximal rate (based on optimizing possibly the pessimistic upper bounds on the discretization error) is obtained for [Formula: see text], for finite volume convergence rate of [Formula: see text]. We conclude with numerical experiments.

A Monte Carlo-Finite Difference Method for Coupled Radiation-Conduction Heat Transfer in Semitransparent Media

Combinaison de la methode de Monte Carlo avec la methode des differences finies pour calculer le transfert de chaleur par rayonnement et conduction thermique en regimes transitoire et permanent dans une couche plane de solide gris homogene