  We study a stochastic phase-field model for tumor growth dynamics coupling a
stochastic Cahn-Hilliard equation for the tumor phase parameter with a
stochastic reaction-diffusion equation governing the nutrient proportion. We
prove strong well-posedness of the system in a general framework through
monotonicity and stochastic compactness arguments. We introduce then suitable
controls representing the concentration of cytotoxic drugs administered in
medical treatment and we analyze a related optimal control problem. We derive
existence of an optimal strategy and deduce first-order necessary optimality
conditions by studying the corresponding linearized system and the backward
adjoint system.
