Variational Methods for Moments of Solutions to Stochastic Differential Equations

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 L\ue9vy 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