Weekly Seminar
Numerical approximation of a stochastic transport problem with Lévy noise
Andreas Stein (University of Stuttgart (Germany))
Wed, 10 Jul 2019 • 15:00-16:00h • Pontdriesch 14-16, Room 008 (SeMath)


To model the dynamics of interest rate and energy forward markets, linear hyperbolic stochastic partial differential equations (SPDEs) may be utilized. The forward rate is then given as the solution to a transport equation with a space-time stochastic process as driving noise. In order to capture temporal discontinuities and allow for heavy-tailed distributions, we consider Hilbert space valued-Lévy processes (or Lévy fields) as driving noise terms. The numerical discretization of the corresponding SPDE involves several difficulties: Low spatial and temporal regularity of the solution to the problem entails slow convergence rates and instabilities for space/time-discretization schemes. Furthermore, the Lévy noise process admits values in a possibly infinite-dimensional Hilbert space H, hence projections into a finite-dimensional subspace HN ⊂ H for each discrete point in time are necessary. Finally, unbiased sampling from the resulting HN-valued Lévy field may not be possible. We introduce a fully discrete approximation scheme that addresses the above issues. A discontinuous Galerkin approach for the spatial approximation is coupled with a suitable time stepping scheme to avoid numerical oscillations and increase the temporal order of convergence. Moreover, we approximate the driving noise process by truncated Karhunen-Loève expansions. The latter essentially yields a sum of scaled and uncorrelated one-dimensional Lévy processes, which may be simulated with controlled bias by Fourier inversion techniques. This is joint work with Andrea Barth (SimTech, University of Stuttgart)