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In this paper, we consider the mixed optimal control of a linear stochasticsystem with a quadratic cost functional, with two controllers-one can chooseonly deterministic time functions, called the deterministic controller, whilethe other can choose adapted random processes, called the random controller.The optimal control is shown to exist under suitable assumptions. The optimalcontrol is characterized via a system of fully coupled forward-backwardstochastic differential equations (FB-SDEs) of mean-field type. We solve theFBSDEs via solutions of two (but decoupled) Riccati equations, and give therespective optimal feedback law for both determinis-tic and random controllers,using solutions of both Riccati equations. The optimal state satisfies a linearstochastic differential equation (SDE) of mean-field type. Both the singularand infinite time-horizonal cases are also addressed.

Mixed Deterministic and Random Optimal Control of Linear Stochastic Systems with Quadratic Costs