Maximum principle for the optimal control problem of a fully coupled stochastic system with state constraints (Q2720036)

The authors consider the optimal problem related to fully coupled forward-backward stochastic differential equations under state constraint described as follows: NEWLINE\[NEWLINEdx(t)=f\bigl( t,x(t),y(t), z(t),v(t) \bigr)dt+ \sigma\bigl(t,x(t), y(t),z(t), v(t)\bigr)dB_t;NEWLINE\]NEWLINE NEWLINE\[NEWLINE-dy(t)=g\bigl(t,x (t),y(t), z(t),v(t)\bigr) dt-z(t)dB_t, \bigl(t\in[0,T] \bigr);NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(0)= x_0;\;y (T)=\varphi\bigl(x(T)\bigr);\;IEG_1\bigl(x(T)\bigr)=0,\text{ and } IEG_0\bigl( y (0)\bigr)=0,NEWLINE\]NEWLINE where all variable are of appropriate dimensions, the permissible domain for the control \(v(t)\) is assumed to be convex and the optimal control is defined under a minimal criterion of the functional \(J(v(\cdot))\) [for the problem and the detailed meanings of notations, cf., \textit{S. Peng} and \textit{Z. Wu}, SIAM J. Control Optimization 37, 825-843 (1999; Zbl 0931.60048)]. Via introducing the related variational equation and using the Ekeland variational principle, the maximum principle, i.e. the necessary condition for the optimal control of the above problem, is then obtained.