Georgios E. Zouraris

Thomas Björk, Anders Szepessy, Raul Tempone et al.

We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on \Ito stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.

Georgios T. Kossioris, Georgios E. Zouraris

We consider an initial- and Dirichlet boundary- value problem for a linear Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we propose an approximate regularized stochastic parabolic problem discretizing the noise using linear splines. Then fully-discrete approximations to the solution of the regularized problem are constructed using, for the discretization in space, a Galerkin finite element method based on H2-piecewise polynomials, and, for time-stepping, the Backward Euler method. Finally, we derive strong a priori estimates for the modeling error and for the numerical approximation error to the solution of the regularized problem.

Georgios E. Zouraris

The solution to the initial and Dirichlet boundary value problem for a semilinear, one dimensional heat equation is approximated by a numerical method that combines the Besse relaxation scheme in time (C. R. Acad. Sci. Paris S{é}r. I, vol. 326 (1998)) with a central finite difference method in space. A new, composite stability argument is developed, leading to an optimal, second-order error estimate in the discrete $L_t^{\infty}(H_x^1)-$norm. It is the first time in the literature where an error estimate for fully discrete approximations based on the Besse relaxation scheme is provided.

Georgios E. Zouraris

We consider a model initial- and Dirichlet boundary- value problem for a fourth-order linear stochastic parabolic equation, in one space dimension, forced by an additive space-time white noise. First, we approximate its solution by the solution of an auxiliary fourth-order stochastic parabolic problem with additive, finite dimensional, spectral-type stochastic load. Then, fully-discrete approximations of the solution to the approximate problem are constructed by using, for the discretization in space, a standard Galerkin finite element method based on $H^2$-piecewise polynomials, and, for time-stepping, the Crank-Nicolson method. Analyzing the convergence of the proposed discretization approach, we derive strong error estimates which show that the order of strong convergence of the Crank-Nicolson finite element method is equal to that reported in [Kosioris and Zouraris MMAN 44 (2010)] for the Backward Euler finite element method.

Georgios E. Zouraris

We formulate an initial- and Dirichlet boundary- value problem for a linear stochastic heat equation, in one space dimension, forced by an additive space-time white noise. First, we approximate the mild solution to the problem by the solution of the regularized second-order linear stochastic parabolic problem with random forcing proposed by Allen, Novosel and Zhang (Stochastics Stochastics Rep., 64, 1998). Then, we construct numerical approximations of the solution to the regularized problem by combining the Crank-Nicolson method in time with a standard Galerkin finite element method in space. We derive strong a priori estimates of the modeling error made in approximating the mild solution to the problem by the solution to the regularized problem, and of the numerical approximation error of the Crank-Nicolson finite element method.

Georgios E. Zouraris

We consider a model initial- and Dirichlet boundary- value problem for a linearized Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we introduce a canvas problem the solution to which is a regular approximation of the mild solution to the problem and depends on a finite number of random variables. Then, fully-discrete approximations of the solution to the canvas problem are constructed using, for discretization in space, a Galerkin finite element method based on $H^2$ piecewise polynomials, and, for time-stepping, an implicit/explicit method. Finally, we derive a strong a priori estimate of the error approximating the mild solution to the problem by the canvas problem solution, and of the numerical approximation error of the solution to the canvas problem.

Georgios E. Zouraris

A linear implicit finite difference method is proposed for the approximation of the solution to a periodic, initial value problem for a Schrodinger-Hirota equation. Optimal, second order convergence in the discrete $H^1-$norm is proved, assuming that $τ$, $h$ and $τ^4/h$ are sufficiently small, where $τ$ is the time-step and $h$ is the space mesh-size. The efficiency of the proposed method is verified by results from numerical experiments.

Georgios T. Kossioris, Georgios E. Zouraris

We consider an initial- and Dirichlet boundary- value problem for a fourth-order linear stochastic parabolic equation, in two or three space dimensions, forced by an additive space-time white noise. Discretizing the space-time white noise a modeling error is introduced and a regularized fourth-order linear stochastic parabolic problem is obtained. Fully-discrete approximations to the solution of the regularized problem are constructed by using, for discretization in space, a standard Galerkin finite element method based on C1 piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates for the modeling error and for the approximation error to the solution of the regularized problem.