This paper continues the study of mean field games introduced in a previous note. It considers the case of Nash equilibria for stochastic control problems with finite horizon. General existence and uniqueness results for the resulting system of partial differential equations are presented. The paper also provides an interpretation of these systems in terms of optimal control. The system of equations is given by: $$ \frac{\partial v}{\partial t}-\nu\Delta v+H(x,\nabla v)=V[m],\qquad v|_{t=0}=v_{0}\big[m(0)\big], $$ $$ \frac{\partial m}{\partial t}+\nu\Delta m+\mathrm{d i v}\bigg(\frac{\partial H}{\partial p}(x,\nabla v)m\bigg)=0,\qquad m|_{t=T}=m_{0} $$ where $ \nu > 0 $, $ m_{0} $ is a given positive bounded smooth function over $ Q = [0, 1]^{d} $, $ \int_{Q} m_{0} = 1 $, all functions are periodic in each $ x_{i} $, H is a smooth convex Hamiltonian, and $ v_{0}[m] $, V[m] are operators that are bounded from $ C^{k,\alpha} $ into $ C^{k+1,\alpha} $. The paper proves the existence of at least one smooth solution (v, m) of the system and, under additional conditions, the uniqueness of the solution. It also discusses various extensions and variants of the system and presents observations on instability and nonuniqueness phenomena. The paper also provides an interpretation of the system in terms of optimal control of partial differential equations. The results are presented in the context of mean field games, which model the behavior of a large number of interacting agents in a system. The paper also discusses the connection between mean field games and optimal control, and provides a rigorous justification of mean field models in the context of stochastic control and Nash equilibria. The paper concludes with a discussion of future research directions, including the possibility of incorporating additional terms into the equations and the study of asymptotic behavior of Nash equilibria.This paper continues the study of mean field games introduced in a previous note. It considers the case of Nash equilibria for stochastic control problems with finite horizon. General existence and uniqueness results for the resulting system of partial differential equations are presented. The paper also provides an interpretation of these systems in terms of optimal control. The system of equations is given by: $$ \frac{\partial v}{\partial t}-\nu\Delta v+H(x,\nabla v)=V[m],\qquad v|_{t=0}=v_{0}\big[m(0)\big], $$ $$ \frac{\partial m}{\partial t}+\nu\Delta m+\mathrm{d i v}\bigg(\frac{\partial H}{\partial p}(x,\nabla v)m\bigg)=0,\qquad m|_{t=T}=m_{0} $$ where $ \nu > 0 $, $ m_{0} $ is a given positive bounded smooth function over $ Q = [0, 1]^{d} $, $ \int_{Q} m_{0} = 1 $, all functions are periodic in each $ x_{i} $, H is a smooth convex Hamiltonian, and $ v_{0}[m] $, V[m] are operators that are bounded from $ C^{k,\alpha} $ into $ C^{k+1,\alpha} $. The paper proves the existence of at least one smooth solution (v, m) of the system and, under additional conditions, the uniqueness of the solution. It also discusses various extensions and variants of the system and presents observations on instability and nonuniqueness phenomena. The paper also provides an interpretation of the system in terms of optimal control of partial differential equations. The results are presented in the context of mean field games, which model the behavior of a large number of interacting agents in a system. The paper also discusses the connection between mean field games and optimal control, and provides a rigorous justification of mean field models in the context of stochastic control and Nash equilibria. The paper concludes with a discussion of future research directions, including the possibility of incorporating additional terms into the equations and the study of asymptotic behavior of Nash equilibria.