Telescoping Approximations for Stochastic Evolution Equations and Randomized Systems

We extend the telescoping approximation framework developed for deterministic constants,functions, differential equations, integral operators, and partial differential equations to stochasticevolution equations and randomized operators. Focusing on mild formulations driven bylinear semigroups and cylindrical Wiener processes, we construct telescoping approximants forstochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs).Under standard assumptions of coercivity and bounded diffusion, we prove that the telescopingstructure propagates through stochastic integration in expectation and in mean square.Specifically, we establish that when the underlying deterministic semigroup approximant hasorder k (meaning operator differences decay as O(n−(k+1))), the mean-square differences of thestochastic solutions decay as E∥Un+1−Un∥2 = O(n−2(k+1)), leading to strong convergence ratesof O(n−(k+1/2)) in the L2(Ω;H) norm. We further show that the increment bounds requiredin Multilevel Monte Carlo (MLMC) follow directly from our telescoping increment frameworkonce a deterministic semigroup approximant is available. Numerical experiments on the Ornstein–Uhlenbeck process and the stochastic heat equation corroborate the predicted convergencerates.