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August 2020 Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations
Benjamin Seeger
Ann. Appl. Probab. 30(4): 1784-1823 (August 2020). DOI: 10.1214/19-AAP1543

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 nontrivial 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.

Citation

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Benjamin Seeger. "Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations." Ann. Appl. Probab. 30 (4) 1784 - 1823, August 2020. https://doi.org/10.1214/19-AAP1543

Information

Received: 1 June 2018; Revised: 1 August 2019; Published: August 2020
First available in Project Euclid: 4 August 2020

MathSciNet: MR4133383
Digital Object Identifier: 10.1214/19-AAP1543

Subjects:
Primary: 35D40, 35K55, 35R60, 60H15, 65M06, 65M12

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 4 • August 2020
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