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This paper is concerned with an optimal control problem for a mean-field linear stochastic differential equation with a quadratic functional in the infinite time horizon. Under suitable conditions, including the stabilizability, the (strong) exponential, integral, and mean-square turnpike properties for the optimal pair are established. The keys are to correctly formulate the corresponding static optimization problem and find the equations determining the correction processes. These have revealed the main feature of the stochastic problems which are significantly different from the deterministic version of the theory. 

The spike variation technique plays a crucial role in deriving Pontryagin's type maximum principle of optimal controls for ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), and (deterministic forward) Volterra integral equations (FVIEs), when the control domains are not assumed to be convex. It is natural to expect that such a technique could be extended to the case of (forward) stochastic Volterra integral equations (FSVIEs). However, by mimicking the case of SDEs, one encounters an essential difficulty of handling an involved quadratic term. To overcome this difficulty, we introduce an auxiliary process for which one can use It\^o's formula, and develop new technologies inspired by stochastic linear-quadratic optimal control problems. Then the suitable representation of the above-mentioned quadratic form is obtained, and the second-order adjoint equations are derived. Consequently, the maximum principle of Pontryagin type is established. Some relevant extensions are investigated as well. 

We investigate a linear–quadratic stochastic zero-sum game where two players lobby a political representative to invest in a wind farm. Players are time-inconsistent because they discount the utility with a non-constant rate. Our objective is to identify a consistent planning equilibrium in which the players are aware of their inconsistency and cannot commit to a lobbying policy. We analyse equilibrium behaviour in both single-player and two-player cases and compare the behaviours of the game under constant and variable discount rates. The equilibrium behaviour is provided in closed-loop form, either analytically or via numerical approximation. Our numerical analysis of the equilibrium reveals that strategic behaviour leads to more intense lobbying without resulting in overshooting. 

A linear-quadratic optimal control problem for a forward stochastic Volterra integral equation (FSVIE) is considered. Under the usual convexity conditions, open-loop optimal control exists, which can be characterized by the optimality system, a coupled system of an FSVIE and a type-II backward SVIE (BSVIE). To obtain a causal state feedback representation for the open-loop optimal control, a path-dependent Riccati equation for an operator-valued function is introduced, via which the optimality system can be decoupled. In the process of decoupling, a type-III BSVIE is introduced whose adapted solution can be used to represent the adapted M-solution of the corresponding type-II BSVIE. Under certain conditions, it is proved that the path-dependent Riccati equation admits a unique solution, which means that the decoupling field for the optimality system is found. Therefore, a causal state feedback representation of the open-loop optimal control is constructed. An additional interesting finding is that when the control only appears in the diffusion term, not in the drift term of the state system, the causal state feedback reduces to a Markovian state feedback. 

In this paper, we present a brief survey for some recent developments of stochastic linear-quadratic optimal controls. We mainly concentrate on the results obtained by the authors and their collaborators in the last decay.