  We propose a time-implicit, finite-element based space-time discretization of
the necessary and sufficient optimality conditions for the stochastic
linear-quadratic optimal control problem with the stochastic heat equation
driven by linear noise of type $[X(t)+\sigma(t)]dW(t)$, and prove optimal
convergence w.r.t. both, space and time discretization parameters. In
particular, we employ the stochastic Riccati equation as a proper analytical
tool to handle the linear noise, and thus extend the applicability of the
earlier work [16], where the error analysis was restricted to additive noise.
