Numerical methods for the 2nd moment of stochastic ODEs

Numerical methods for stochastic ordinary differential equations typically estimate moments of the solution from sampled paths. Instead, in this paper we directly target the deterministic equation satisfied by the first and second moments. For the canonical examples with additive noise (Ornstein–Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic equations in variational form and discuss their well-posedness in detail. Notably, the second moment equation in the multiplicative case is naturally posed on projective– injective tensor products as trial–test spaces. We propose Petrov–Galerkin discretizations based on tensor product piecewise polynomials and analyze their stability and convergence in the natural norms.