Application of Multiple Fourier-Legendre Series to Implementation of Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations

Abstract

The article is devoted to the application of multiple Fourier-Legendre series to implementation of strong exponential Milstein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations with multiplicative trace class noise. These methods have strong orders of convergence $1.0-\epsilon$ and $1.5-\epsilon$ correspondingly (here $\epsilon$ is an arbitrary small positive real number) with respect to the temporal discretization. The theorem on mean-square convergence of approximations of iterated stochastic integrals of multiplicities 1 to 3 with respect to the infinite-dimensional $Q$-Wiener process is formulated and proved. The results of the article can be applied to implementation of exponential Milstein and Wagner-Platen methods for non-commutative semilinear stochastic partial differential equations with multiplicative trace class noise.

arXiv comment32 pages. Some changes along the text in the whole. arXiv admin note: substantial text overlap with arXiv:1905.03724

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Cited by in corpus (8)

• Strong Numerical Methods of Orders 2.0, 2.5, and 3.0 for Ito Stochastic Differential Equations Based on the Unified Stochastic Taylor Expansions and Multiple Fourier-Legendre Series
• Application of the Method of Approximation of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series to the High-Order Strong Numerical Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
• Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Iterated Fourier Series Converging Pointwise
• Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in the Mean: General Case of Series Summation
• New Simple Method of Expansion of Iterated Ito Stochastic integrals of Multiplicity 2 Based on Expansion of the Brownian Motion Using Legendre Polynomials and Trigonometric Functions
• Four New Forms of the Taylor-Ito and Taylor-Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochastic Differential Equations