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We introduce a time-implicit, finite-element based space-time discretizationscheme for the backward stochastic heat equation, and for the forward-backwardstochastic heat equation from stochastic optimal control, and prove strongrates of convergence. The fully discrete version of the forward-backwardstochastic heat equation is then used within a gradient descent algorithm toapproximately solve the linear-quadratic control problem for the stochasticheat equation driven by additive noise.

Strong Rates of Convergence for Space-Time Discretization of the  Backward Stochastic Heat Equation, and of a Linear-Quadratic Control Problem  for the Stochastic Heat Equation