A Second-order Stochastic Maximum Principle For Singular Mean-field Optimal Stochastic Controls

In this paper, we study the mean-field singular optimal control problem for stochastic differential equations. Pardoux and Peng [1] firstly introduced the non-linear backward stochastic differential equations in 1990. In the same year, Peng [4] found the first-order maximum principle for classical stochastic control problem. In 2012, Li [8] deduced first-order stochastic maximum principle for mean-field controls.A mean-field admissible control u(·) is called singular on control region V, if V(?)U is nonempty and for a.e.,t∈[0,1], we have:Obviously, the first-order maximum principle doesn’t work in this situation.We consider the state equation as follow:Cost functional is defined as: First, we estimate the solution of state equation, expand the state variation and cost functional with respect to control functions into second order with the help of Taylar expansion. Then, by choosing appropriate fist and second order adjoint processes, we get the variational inequality. Finally, via a generalized spike variation technique together with the vector-valued measure theory, we drive the second-order stochastic maximum principle for mean-field singular controls.(A3) The functionsare Borel measurable with respect to their respective arguments, continuous in u, continuously differentiale in x’,x, for each fixed (t,u), and for constant K0, Moreover, all the derivatives involved above are Borel measurable, and are continuous in x’ x.(A4) The first order derivatives involved above are continuous in u on U. The function f,σ’,l,h, respectively, are continuous second-order derivatives in x’,x. The second-order derivatives are Borel measurable with respect to (t,x’,x,u), and are bounded by the constant K0, that isWe drive the second-order stochastic maximum principle for mean-field singular controls as follows:Assume that (A3) and (A4) are satisfied. Let (y(·),u(·)) be an optimal pair, and u(·) be singular on the control region V. Then, there is a subset I0(?)[0,1] which is full measure, such that at each t∈I0, (y(·),u(·))) satisfies, in addition to the first-order maximum condition, the following second-order maximum condition:In which, H is Hamiltonian, P(t) is second-order adjoint process, andThen, we stuied the linear-quadratic control problem in singular mean-field control.