Variational Methods for Moments of Solutions to Stochastic Differential Equations
Licentiatavhandling, 2016

Numerical methods for stochastic differential equations typically estimate moments of the solution from sampled paths. Instead, we pursue the approach proposed by A. Lang, S. Larsson, and Ch. Schwab [1], who derived well-posed deterministic, tensorized evolution equations for the second moment and the covariance of the solution to a parabolic stochastic partial differential equation driven by additive Wiener noise. In Paper I we consider parabolic stochastic partial differential equations with multiplicative Lévy noise of affine type. For the second moment of the mild solution, a deterministic space-time variational problem is derived. It is posed on projective and injective tensor product spaces as trial and test spaces. Well-posedness is proven under appropriate assumptions on the noise term. From these results, a deterministic equation for the covariance function is deduced. These deterministic equations in variational form are used in Paper II to derive numerical methods for approximating the first and second moment of the solution to a stochastic ordinary differential equation driven by additive or multiplicative Wiener noise. For the canonical examples with additive noise (Ornstein-Uhlenbeck process) and multiplicative noise (geometric Brownian motion) we first recall the variational problems satisfied by the first and the second moments of the solution processes and discuss their well-posedness in detail. For the considered examples, well-posedness beyond the assumptions on the multiplicative noise term made in Paper I are proven. We propose Petrov-Galerkin discretizations based on tensor product piecewise polynomials and analyze their stability and convergence in the natural norms. [1] A. Lang, S. Larsson, and Ch. Schwab. Covariance structure of parabolic stochastic partial differential equations. Stoch. PDE: Anal. Comp., 1(2013), pp. 351-364.

Additive and multiplicative noise

Stochastic partial differential equation

Projective and injective tensor product space

Hilbert tensor product space

Space-time variational problem

Petrov-Galerkin discretization

Stochastic ordinary differential equation

Pascal, Chalmers tvärgata 3, Chalmers.
Opponent: Prof. Dr. Sonja Cox, Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Netherlands.

Författare

Kristin Kirchner

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Kristin Kirchner, Annika Lang, and Stig Larsson. Covariance structure of parabolic stochastic partial differential equations with multiplicative Lévy noise. Preprint, arXiv:1506.00624.

Roman Andreev and Kristin Kirchner. Numerical methods for the 2nd moment of stochastic ODEs. Preprint, arXiv:1611.02164.

Ämneskategorier

Beräkningsmatematik

Sannolikhetsteori och statistik

Utgivare

Chalmers tekniska högskola

Pascal, Chalmers tvärgata 3, Chalmers.

Opponent: Prof. Dr. Sonja Cox, Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Netherlands.

Mer information

Skapat

2016-11-22