Abstract: In the numerical analysis of stochastic ordinary differential equations (SODEs), G. N.

Milstein proposed an important convergence criterion to evaluate the mean-square convergence

order for numerical approximations of SODEs, which is called fundamental convergence theorem. 

Motivated by Milstein’s work,  we proposed the fundamental convergence theorems on the mean-

square convergence orders of numerical approximations for a class of important backward stochastic

differential equations (BSDEs) and for stochastic Schrödinger equation. The theorems  show that the

mean-square order of convergence of a numerical method for BSDEs depends on the properties of

mean-square deviation of one-step approximation only, while the mean-square convergence order

of a numerical method for stochastic Schrödinger equation depends on the properties of one-step

approximation both in mean and mean-square sense, and on the estimate of semigroup operators.