Time-discretised Galerkin approximations of parabolic stochastic PDE's

Abstract

The global discretisation error is estimated for strong time discretisations of finite dimensional Ito stochastic differential equations (SDEs) which are Galerkin approximations of a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues λ1 ≤ λ2 ≤ … in its drift term. If an order γ strong Taylor scheme with time-step δ is applied to the N dimensional Ito-Galerkin SDE, the discretisation error is bounded above bywhere [x] is the integer part of the real number x and the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration.