  We study the adapted solution, numerical methods, and related convergence
analysis for a unified backward stochastic partial differential equation
(B-SPDE). The equation is vector-valued, whose drift and diffusion coefficients
may involve nonlinear and high-order partial differential operators. Under
certain generalized Lipschitz and linear growth conditions, the existence and
uniqueness of adapted solution to the B-SPDE are justified. The methods are
based on completely discrete schemes in terms of both time and space. The
analysis concerning error estimation or rate of convergence of the methods is
conducted. The key of the analysis is to develop new theory for random field
based Malliavin calculus to prove the existence and uniqueness of adapted
solutions to the first-order and second-order Malliavin derivative based
B-SPDEs under random environments.
