2021/07/08 by Vo Anh, Anh, Vo V., Andriy Olenko +3 · 2 citations
Economics, Econometrics and Finance · Mathematics · #33C55 #35R11 #35R60 #60G22 #60G60 #60H15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Financial Risk and Volatility Modeling #Fractional Differential Equations Solutions #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2107.03717
openalex publication_date 2021/07/08 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
This paper develops a fractional stochastic partial differential equation\n(SPDE) to model the evolution of a random tangent vector field on the unit\nsphere. The SPDE is governed by a fractional diffusion operator to model the\nL 'evy-type behaviour of the spatial solution, a fractional derivative in\ntime to depict the intermittency of its temporal solution, and is driven by\nvector-valued fractional Brownian motion on the unit sphere to characterize its\ntemporal long-range dependence. The solution to the SPDE is presented in the\nform of the Karhunen-Lo `eve expansion in terms of vector spherical\nharmonics. Its covariance matrix function is established as a tensor field on\nthe unit sphere that is an expansion of Legendre tensor kernels. Approximations\nto the solutions are studied and convergence rates of the approximation errors\nare given. It is demonstrated how these convergence rates depend on the decay\nof the power spectrum and variances of the fractional Brownian motion.\n