Time-dicretization of a linear Shrödinger equation with additive noise and final value linearly dependent on initial value

We consider a heat equation type equation with presence of a complex unit, here driven by an additive Q-Wiener process with finite trace. The somewhat exotic situation is that here the final value depends linearly on the initial value. Existence and uniqueness of solution together with time discretization algorithms such as exact simulation, exponential integrator and Euler-type schemes is investigated. For implementation, error created from truncation of the Q-Wiener process series representation is considered. Spatial regularity is also an issue.