Some remarks on the Riccati equation arising in an optimal control problem with state- and control-dependent noise

This paper solves a quadratic optimal control for a linear stochastic evolution equation with unbounded coefficients. It is assumed that the stochastic noise depends both on the state and on the control. The dynamic programming approach is used and attention is focused on the Riccati equation. In sections 5 and 6 some attractivity and maximality properties of the solutions of the algebraic Riccati equation are proved and it is shown that, in some special cases, there exists a maximal solution.