# Constrained Dynamic Mean-Variance Portfolio Selection in Continuous-Time
## 2 Problem Formulation  
In this work, we consider a financial market consisting of one risk-free asset and \(n\) risky assets, which can be traded continuously within time horizon \([0,T]\). We assume that all randomness is modeled by a complete filtrated probability space \((\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq 0},\mathbb{P})\), on which \(\mathcal{F}_{t}\)-adapted \(n\)-dimensional Brownian motion \(W(t):=(W^{1}(t),\cdots,W^{n}(t))\) is defined. Denote \(\mathcal{F}_{t}\) as the augmented \(\sigma\)-algebra generated by \(W(t)\), which implies the information set available at time \(t\), \(t\in[0,T]\). \(\mathcal{L}_{\mathcal{F}}^{2}(0,T;\mathbb{R})\) is the set of \(\mathbb{R}\)-valued, \(\mathcal{F}_{t}\)-adapted and square integrable stochastic processes. The price of the risk-free asset is \(S_{0}(t)\), which is governed by the following equation:  
\[ \begin{cases}dS_{0}(t)=r(t)S_{0}(t)dt,\ \ t\in[0,T],\\ S_{0}(0)=s_{0}>0,\end{cases}  \tag{1}\]  
where \(r(t)>0\) is the risk-free return rate. The price of \(n\) risky assets satisfy the SDE:  
\[ \begin{cases}dS_{i}(t)=S_{i}(t)\{u_{i}(t)dt+\sum_{j=1}^{n}\sigma_{ij}(t)dW^{j}( t)\},\\ S_{i}(0)=s_{i}>0\end{cases}  \tag{2}\]  
where \(\mu_{i}(t)>0\) and \(\sigma_{ij}(t)\) are the appreciation rate and volatility coefficient, respectively. Let \(\mu(t):=\big{(}\mu_{1}(t),\cdots,\mu_{n}(t)\big{)}^{\prime}\) and  
\[\sigma(t):=\left[ \begin{array}{ccc}\sigma_{11}(t)&\cdots&\sigma_{1n}(t)\\ \cdots&\cdots&\cdots\\ \sigma_{n1}(t)&\cdots&\sigma_{nn}(t)\end{array} \right],\]  
for all \(t\in[0,T]\). We assume that \(r(t)\), \(\mu(t)\) and \(\sigma(t)\) are the deterministic functions of time and they are bounded for \(t\in[0,T]\). We also assume the following nondegeneracy condition holds for some constant \(\delta>0\),  
\[\sigma(t)\sigma(t)^{\prime}\geq\delta\mathbf{I},\ \ t\in[0,T].\]  
An investor enters the market with initial wealth \(x_{0}\) and allocates this wealth continuously during the invest horizon \([0,T]\). The total wealth at time \(t\) is denoted as \(x(t)\) and the portfolio decision vector is denoted as \(u(t){:=}\big{(}u_{1}(t),\cdots,u_{n}(t)\big{)}^{\prime}\), which represents the allocations of the wealth in \(n\) risky assets. The wealth level \(x(t)\) evolves according to the following SDE (e.g., see [PERSON] et al. [3]):  
\[ \begin{cases}dx(t)=\Big{(}r(t)x(t)+b(t)^{\prime}u(t)\Big{)}dt+u(t)^{\prime} \sigma(t)dW(t),\quad t\in[0,T],\\ x(0)=x_{0}>0,\end{cases}  \tag{3}\]  
where \(b(t):=\big{(}b_{1}(t),\cdots,b_{n}(t)\big{)}^{\prime}=\big{(}\mu_{1}(t)-r(t), \cdots,\mu_{n}(t)-r(t)\big{)}^{\prime}\) is the excess return rate vector. We also assume the following condition holds. **Assumption 1**. : _There exists some \(i^{*}\in\{1,\cdots,n\}\) such that \(b_{i^{*}}(t)>0\)._  
Assumption 1 says that the return rates of some risky assets should be greater than the return rate of the risk-free asset, which is reasonable in real markets. Motivated by the restrictions on real investments, we consider the following constraint for the portfolio decision vector:  
\[H(t)u(t)\geq 0,\ \ t\in[0,T], \tag{4}\]  
where \(H(t)\in\mathbb{R}^{k\times n}\) is the deterministic matrix for \(t\in[0,T]\). The above constraint is known as the convex cone constraint, which includes various portfolio constraints as its special cases, e.g., the no-shorting constraint ([12]). In the following, we use \(\text{Var}[x]{:=}\mathrm{E}[(x-\mathrm{E}[x])^{2}]\) to denote the variance of some random variable \(x\). The investor adopts the following dynamic MV portfolio decision model to guide his investment:  
\[(\mathcal{P}_{\text{MV}}):\ \min_{\{u(t)\}|_{t=0}^{T}} \ \ \text{Var}[x(T)]+\int_{0}^{T}\mathrm{E}[u(t)^{\prime}R(t)u(t)]dt\] \[(s.t.) \ \mathrm{E}[x(T)]=d,\] \[\{x(t),u(t)\}\ \ \text{satisfies (\ref{eq:1}) and (\ref{eq:2})}.\]where \(d\) is expected terminal wealth level. Usually, we set \(d>x_{0}e^{\int_{0}^{T}r(s)ds}\); i.e., the target expected terminal wealth should be greater than that obtained by investing all into the risk-free account.