# Constrained Dynamic Mean-Variance Portfolio Selection in Continuous-Time
## 1 Introduction  
The classical static mean-variance (MV) model was pioneered by [PERSON] [1] more than sixty years ago, which laid the foundation of modern financial theory. However, such a static policy is not a good choice since the investor usually needs to adjust his/her portfolio policy to achieve better performance according to updated information. Due to the non-separability of the variance term in the language of dynamic programming, this problem is not trivial analytically. In 2000, the static MV model was extended to the multi-period setting by [PERSON] and [PERSON] [2] and to the continuous-time setting by [PERSON] and [PERSON] [3] by using the proposed embedding method. From then on, the past decades have witnessed significant progress on the dynamic MV portfolio selection analysis by leaps and bounds; see, for example, [PERSON] et al. [4], [PERSON] et al. [5], [PERSON] et al. [6], [PERSON] et al. [7], [PERSON] et al. [8], [PERSON] et al. [9], [PERSON] et al. [10] and [PERSON] et al. [11]. One prominent attraction of the dynamic MV portfolio selection models is their explicit portfolio policy, which can be derived by using the embedding method and dynamic programming approach. However, in real portfolio management, some constraints on portfolio strategy are inevitable; e.g., the investor is usually subject to some limits of consideration of the risk or economic regulations. Generally speaking, when we consider the strategy constraints, they usually destroy the advantages of the MV model. That is to say, under such cases, it is hard to obtain the closed form of a portfolio policy except in a few special cases. To address this issue, [PERSON] et al. [12] adopted the viscosity solution of the partial differential equation and successfully characterized the analytical solution of the continuous-time MV portfolio selection problem with no-shorting constraints. [PERSON] et al. [13] considered the discrete-time version of this type of problem with no-shorting constraints and solved it by using dynamic Programming. [PERSON] et al. [14] developed the optimal portfolio policy for the dynamic MV portfolio selection model with a cardinality constraint with respect tothe active periods of time. [PERSON] et al. (2015) investigated the stochastic linear-quadratic (LQ) optimal control problem with the linear control constraints, which can be regarded as a generalization of dynamic MV portfolio selection with no-shorting constraints. In addition, some promising results have emerged recently for dynamic MV portfolio selection with some other constraints, such as no bankruptcy constraints (see, e.g., [PERSON] et al. (2018) and [PERSON] et al. (2018)). This paper studies the dynamic MV portfolio selection problem with cone constraints in continuous-time. As for the discrete-time versions of problems of this type with cone constraints, [PERSON] et al. (2018) investigated them in detail and solved them by using dynamic programming. The revealed optimal portfolio policy is a piece-wise affine function of current wealth level and can be computed efficiently by solving two important equations offline. [PERSON] et al. (2015) further verified this significant result by utilizing the state separation property induced from its structure. As for the continuous-time versions of problems of this type, [PERSON] and [PERSON] (2017) solved them by using the backward stochastic differential equation approach. More specifically, they introduced two extended stochastic Riccati equations (ESREs) and characterized the optimal portfolio policy by using the solutions of these two ESREs. However, their work did not demonstrate how to construct these two ESREs by using the special structure of a model rather than by suspecting them. Generally speaking, this task has not been accomplished. Recently, [PERSON] et al. (2018) proposed an important state separation theorem and answered this question under the framework of the stochastic LQ control problem with linear control constraints.