[Paper Review] Numerical methods for the deterministic second moment equation of parabolic stochastic PDEs

Kirchner, Kristin|arXiv (Cornell University)|Nov 7, 2016

Stochastic processes and financial applications15 references1 citations

TL;DR

This paper proposes direct numerical solution of deterministic variational equations for the second moment of parabolic stochastic PDEs with affine multiplicative Lévy noise, using Petrov–Galerkin discretizations on tensor product spaces. It establishes well-posedness of the second moment equation under less restrictive conditions than prior work and proves stability and quasi-optimal convergence for space-time conforming schemes, validated by numerical experiments showing first-order convergence in time.

ABSTRACT

Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the first and second moments, as well as the covariance. In the first part, we focus on stochastic ordinary differential equations. For the canonical examples with additive noise (Ornstein-Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic equations in variational form and discuss their well-posedness in detail. Notably, the second moment equation in the multiplicative case is naturally posed on projective-injective tensor product spaces as trial-test spaces. We construct Petrov-Galerkin discretizations based on tensor product piecewise polynomials and analyze their stability and convergence in these natural norms. In the second part, we proceed with parabolic stochastic partial differential equations with affine multiplicative noise. We prove well-posedness of the deterministic variational problem for the second moment, improving an earlier result. We then propose conforming space-time Petrov-Galerkin discretizations, which we show to be stable and quasi-optimal. In both parts, the outcomes are illustrated by numerical examples.

Motivation & Objective

• To derive and analyze deterministic variational equations for the first and second moments of solutions to stochastic ODEs and PDEs with multiplicative noise.
• To overcome limitations in prior work by proving well-posedness of the second moment equation without requiring small noise intensity.
• To develop stable and convergent space-time Petrov–Galerkin discretizations for the second moment equation in the multiplicative noise case.
• To extend results from stochastic ODEs to parabolic stochastic PDEs with affine multiplicative Lévy noise using semigroup theory on tensor product spaces.
• To validate the proposed methods through numerical experiments demonstrating convergence and stability.

Proposed method

• Derive variational forms for the first and second moments of stochastic ODEs with additive and multiplicative Wiener noise, using functional analytic tools and tensor product spaces.
• Prove well-posedness of the second moment equation in the multiplicative case by analyzing the trace product and using projective–injective tensor product spaces as trial and test spaces.
• Construct Petrov–Galerkin discretizations using tensor product piecewise polynomials in space and time, with special treatment for the trace product in the multiplicative case.
• Introduce postprocessing and modified trace product formulations (e.g., iE⋆₂/Q) to restore stability and consistency in the multiplicative case.
• Generalize the framework to parabolic SPDEs with affine multiplicative Lévy noise using C₀-semigroups on projective tensor product spaces.
• Establish quasi-optimality and stability for space-time conforming schemes, particularly for the CN∗₂ time discretization, avoiding postprocessing.

Experimental results

Research questions

• RQ1 Can the second moment of a parabolic SPDE with multiplicative noise be solved via a deterministic variational formulation without requiring small noise intensity?
• RQ2 How can stable and convergent space-time Petrov–Galerkin discretizations be constructed for the second moment equation in the multiplicative noise case?
• RQ3 What is the role of tensor product spaces—particularly projective–injective and Hilbert tensor products—in ensuring well-posedness and stability of the second moment equation?
• RQ4 How do different time discretization schemes (e.g., iE⋆, CN∗₂) compare in stability and convergence for the second moment problem?
• RQ5 Can the numerical framework be extended from stochastic ODEs to parabolic SPDEs with affine multiplicative Lévy noise while preserving stability and convergence?

Key findings

• The second moment equation for multiplicative noise is well-posed on projective–injective tensor product spaces, with well-posedness established under the condition CG = (108) < ∞, which is less restrictive than the smallness assumption in prior work.
• The Petrov–Galerkin discretization based on the CN∗₂ time scheme achieves stability and quasi-optimal convergence without requiring postprocessing, unlike lower-order schemes.
• Numerical experiments confirm first-order convergence in the temporal discretization parameter k for both diagonal coefficients and total error, in agreement with Theorem 4.6.
• The use of symmetrization and preconditioning in conjugate gradient solvers enables efficient solution of the large-scale discrete systems arising from the tensor product discretization.
• The framework generalizes to vector-valued problems via C₀-semigroups on projective tensor product spaces, enabling analysis of SPDEs with multiplicative Lévy noise.
• The proposed method avoids Monte Carlo sampling by directly solving deterministic moment equations, offering potential for reduced computational cost and memory usage through compressive space-time schemes.