Optimal control problem for nonlinear stochastic difference second kind Volterra equations (Q1376699)

The author considers the optimal stochastic control problem for the discrete-time system \[ x(i+1) =\eta (i+1) +\Phi (i+1, x_{i+1}) +\sum^i_{j=0} a\bigl(i,j,x_j,u(j) \bigr)+ \sum ^i_{j=0} b\bigl(i,j,x_j, u(j)\bigr) \xi(j), \tag{1} \] \[ x(0)= \eta(0) \] and the cost functional \[ J(u)= E\Biggl[F(x_N) +\sum^{N-1}_{j=0} G\bigl(j,x_j, u(j)\bigr) \Biggr].\tag{2} \] Under suitable assumptions, necessary optimality conditions are derived. As an example a linear-square optimal control problem is considered.