Maximum principle for an optimal control problem associated to a SPDE with nonlinear boundary conditions

We study a control problem where the state equation is a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise. We allow the control to act on the boundary and set boundary conditions which result in a stochastic differential equation for the trace of the solution on the boundary. This work provides necessary and sufficient conditions of optimality in the form of a maximum principle. We also provide a result of existence for the optimal control in the case where the control acts linearly.