2021/05/19 by Aurélien Deya, Deya, Aurélien, Renaud Marty +1
Decision Sciences · Economics, Econometrics and Finance · Engineering · #FOS: Mathematics #Probabilistic and Robust Engineering Design #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications
paper · doi:10.48550/arxiv.2105.08977
openalex publication_date 2021/05/19 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We study a full discretization scheme for the stochastic linear heat equation \begincases∂t ⟨Ψ⟩ = Δ⟨Ψ⟩ +B , t∈ [0,1], x∈ ℝ,
⟨Ψ⟩0=0 ,\endcases when B is a very rough space-time fractional noise. The discretization procedure is divised into three steps: (i) regularization of the noise through a mollifying-type approach; (ii) discretization of the (smoothened) noise as a finite sum of Gaussian variables over rectangles in [0,1]× ℝ; (iii) discretization of the heat operator on the (non-compact) domain [0,1]× ℝ, along the principles of Galerkin finite elements method. We establish the convergence of the resulting approximation to ⟨Ψ⟩, which, in such a specific rough framework, can only hold in a space of distributions. We also provide some partial simulations of the algorithm.