As we have seen in Chap. 4 , there exists a degree of freedom in choosing a stabilizing control law in a state feedback form for a controlled jump linear stochastic system. This freedom may be used in order to improve some suitable performance specifications. A natural performance criterion is described by a trade off between the cost of the deviation of an output from a required value and the amount of energy spent by the controller. Roughly speaking, this idea leads to a class of quadratic cost function of the form \(\begin{aligned} J(u)=\int \limits _{0}^{\infty }\mathbb {E}[|z(t)-z_d(t)|^2]\text {d}t+\gamma ^2\int \limits _{0}^{\infty }\mathbb {E}[|u(t)|^2]\text {d}t. \end{aligned}\) The term \(\int \nolimits _{0}^{\infty }\mathbb {E}[|z(t)-z_d(t)|^2]\text {d}t\) measures the cost of the deviation of the output \(z(\cdot )\) from a desired value \(z_d(\cdot )\) and \(\int \nolimits _{0}^{\infty }\mathbb {E}[|u(t)|^2]\text {d}t\) measures the control effort. The coefficient \(\gamma ^2\) reflects the trade off between the two terms. In the present chapter we will address the problem of the design of a control law, in a state feedback form, which minimizes a wide class of quadratic cost functions. The minimization of the cost function is done along the state trajectories of a control jump linear stochastic system. Both the finite time horizon case as well as the infinite time horizon case, are considered. Before solving such optimal control problems we first give a fundamental result in Sect. 6.1 where we show that the considered quadratic cost functions can be expressed equivalently in terms of a solution of an adequately defined stochastic jump matrix Riccati differential equation (See Lemma 6.1). This equivalent version of the cost functions will be helpful in the derivation of the explicit formulae of the optimal controls. In Sect. 6.2, the finite time horizon case is addressed. We show that the feedback gain corresponding to the optimal controller is constructed based on the solution of an adequately defined stochastic jump matrix Riccati differential equation. Such a solution must satisfy a terminal value condition as well as the sign conditions on the quadratic terms of the aforementioned Riccati differential equation (see Theorem 6.1). Sections 6.3 and 6.4 are devoted to the infinite time horizon case. Two different cases will be analyzed: We finally apply the obtained results to the case of sampled-data linear stochastic systems (Sect. 6.5).