Weak convergence of the Milstein scheme for semi-linear parabolic stochastic evolution equations

Abstract

The numerical analysis of the Milstein scheme for stochastic ordinary differential equations (SDEs) is relatively well understood. It converges with both strong and weak order one. However, much less is known about the Milstein scheme and its variants when applied to stochastic partial differential equations or more general stochastic evolution equations. This thesis focuses on the weak convergence of the Milstein scheme in the latter setting. We prove that, similar to the SDE case, it also achieves an order of almost one — specifically, an order of 1 − ε for all ε > 0. More concretely, we work in the semigroup framework introduced by Da Prato and Zabczyk and examine the approximation of mild solutions of equations of semi-linear parabolic type. In addition, we allow the drift coefficient of the evolution equation to take values in certain distribution spaces associated to the dominating linear operator. In that case, the order of convergence depends on the regularity of the coefficients and tends to zero as the regularity decreases. The proof employs elements of the mild stochastic calculus recently introduced by Da Prato, Jentzen and Röckner (Trans. Amer. Math. Soc., 372(6), 2019) and crucially depends on recent results on the regularity of solutions to the associated infinite-dimensional Kolmogorov backward equation by Andersson, Hefter, Jentzen and Kurniawan (Potential Anal., 50(3), 2019). It is based on work by Jentzen and Kurniawan investigating Euler-type schemes (Found. Comput. Math., 21(2), 2021).