Weak convergence rates for temporal numerical approximations of the semilinear stochastic wave equation with multiplicative noise

In this work we establish weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise, in particular, for the hyperbolic Anderson model. For this class of stochastic partial differential equations the weak convergence rates we obtain are indeed twice the known strong rates. To the best of our knowledge, our findings are the first in the scientific literature which provide essentially sharp weak convergence rates for temporal discretisations of stochastic wave equations with multiplicative noise. Key ideas of our proof are a sophisticated splitting of the error and applications of the recently introduced mild Itô formula. We complement our analytical findings by means of numerical simulations in Python for the decay of the weak approximation error for SPDEs for four different test functions.