The Mean-field Stochastic Maximum Principle Of The General Case

The paper mainly studies two kinds of stochastic control problems of mean-field type.The coefficients of state equations not only depend on the solution but also the solution's law.And the cost functional is also mean-field type.Firstly,we study the state equation as follows which is a mean-field stochastic differential equation:The cost functional is as follows(2)The derivatives of b,? in(x,?,v)are Lipschitz continuous and bounded.(3)The derivatives of h,? in(x,?,v)and(x,?)are Lipschitz continuous and bound-ed by C(1 + |x| + |v|)and C(1 + |x|).(4)b,? are Lipschitz continuous and linear growth with respect to(x,?)and uniformly with respect to v.(H3.2)H(x,?,p,q,v)is convex in v.The control space is convex.We study the conditions that the stochastic optimal control satisfied.Using convex disturbance and duality we get the necessary conditions of optimal control,and we also get the sufficient conditions.In the end,we study the state equation which is a decoupled mean-field forward-backward stochastic differential equation:where b,?,f,? are the given mappings,and the initial value ? is a F0 measurable random variable.We assume:(2)The derivatives of b,?,f in(x,?,u)and(x,y,z,v,u)are Lipschitz continuous and bounded.(3)The derivatives of ? in(x,?)are Lipschitz continuous and bounded by C(1 + ?|x|).(4)For given control u,f(·,0,0,?0?u)?HF2(0,T;Rm).(5)b,? are Lipschitz continuous and linear growth with respect to(x,?),f is uniformly Lipschitz with respect to(x,y,z,v).All are uniformly with respect to u.The cost functional is Similarly,we use convex disturbance and duality to get the necessary conditions for optimal control.