Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations

TL;DR

This paper introduces a novel regularization approach for constructing approximation schemes for viscosity solutions of fully nonlinear stochastic PDEs, ensuring convergence despite irregular time dependence, with applications to equations driven by white noise.

Contribution

The paper develops a new regularization method for pathwise stochastic PDEs that guarantees monotonicity and convergence, extending classical viscosity theory to irregular paths.

Findings

Established error estimates depending on path modulus of continuity.

Proved convergence with probability one for equations with multiplicative white noise.

Demonstrated convergence in distribution using scaled random walks.

Abstract

The aim of this paper is to develop a general method for constructing approximation schemes for viscosity solutions of fully nonlinear pathwise stochastic partial differential equations, and for proving their convergence. Our results apply to approximations such as explicit finite difference schemes and Trotter-Kato type mixing formulas. The irregular time dependence disrupts the usual methods from the classical viscosity theory for creating schemes that are both monotone and convergent, an obstacle that cannot be overcome by incorporating higher order correction terms, as is done for numerical approximations of stochastic or rough ordinary differential equations. The novelty here is to regularize those driving paths with non-trivial quadratic variation in order to guarantee both monotonicity and convergence. We present qualitative and quantitative results, the former covering a wide variety of schemes for second-order equations. An error estimate is established in the Hamilton-Jacobi case, its merit being that it depends on the path only through the modulus of continuity, and not on the derivatives or total variation. As a result, it is possible to choose a regularization of the path so as to obtain efficient rates of convergence. This is demonstrated in the specific setting of equations with multiplicative white noise in time, in which case the convergence holds with probability one. We also present an example using scaled random walks that exhibits convergence in distribution.