Boris Baeumer: Approximations of Lévy Processes on Bounded Domains

Abstract: A numerical approximation to the solution of the governing equation for a Lévy process reveals how to impose boundary conditions and how these boundary conditions modify the original process. It is a nice example on how numerics can not only provide intuition and circumstantial evidence but understanding and proof. In the special case of a stable process, a process governed by a fractional derivative, the first eigenmode for Dirichlet boundary conditions behaves like xa-1 at the boundary, destroying first order convergence of commonly used approximation schemes for the alpha-derivative operator, like the Grunwald approximation. This led to the discovery of a new scheme that is positivity preserving and of order alpha, involving the polylogarithm and Riemann Zeta function.


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