Strong Convergence of the Linear Implicit Euler Method for the Finite\n Element Discretization of Semilinear non-Autonomous SPDEs Driven by\n Multiplicative or Additive Noise

2019/01/08 by Jean Daniel Mukam, Mukam, Jean Daniel, Antoine Tambue +1
Economics, Econometrics and Finance · Social Sciences · #FOS: Mathematics #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Numerical Analysis (math.NA) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1901.03189

openalex publication_date 2019/01/08 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28

Abstract

This paper aims to investigate the numerical approximation of semilinear\nnon-autonomous stochastic partial differential equations (SPDEs) driven by\nmultiplicative or additive noise. Such equations are more realistic than\nautonomous SPDEs while modeling real world phenomena. Numerical approximations\nfor autonomous SPDEs are thoroughly investigated in the literature, while the\nnon-autonomous case is not yet well understood. The non-autonomous SPDE is\ndiscretized in space by the finite element method and in time by the linear\nimplicit Euler method. We break the complexity in the analysis of the time\ndepending, not necessarily self-adjoint linear operators with the corresponding\nsemi group and provide the strong convergence result of the fully discrete\nscheme toward the exact solution in the root-mean-square L² norm. The\nresults indicate how the converge order depends on the regularity of the\ninitial solution and the noise. In particular, for multiplicative trace class\nnoise we achieve convergence order \O(h^2-\ε+\Δ t^1/2)\nand for additive noise with trace class, we achieve convergence order\n\O(h^2-\ε+\Δ t^1-\ε), for an arbitrarily small\n\ε>0. Numerical experiments to sustain our theoretical results are\nprovided.\n

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