General Presentation of Mean Field Control Problems

Consider a probability space \((\Omega,\mathcal{A},P)\) and a filtration \({\mathcal{F}}^{t}\) generated by a n-dimensional standard Wiener process w(t). The state space is \({\mathbb{R}}^{n}\) with the generic notation x and the control space is \({\mathbb{R}}^{d}\) with generic notation v.