# Constrained Dynamic Mean-Variance Portfolio Selection in Continuous-Time
## 4 An Alternative Method for Problem (\(\mathcal{P}_{\text{MV}}\)) without the Penalty Term  
In this section, we provide an alternative method to resolve this dynamic MV portfolio selection problem with cone constraints. Consider the following portfolio selection problem (\(\mathcal{A}_{\text{MV}}\)) without the penalty term \(u(t)^{\prime}R(t)u(t)\); i.e.,  
\[(\mathcal{A}_{\text{MV}}): \min_{\{u(t)\}_{t=0}^{T}}\quad\text{Var}[x(T)]\] \[\quad\quad(s.t): \text{E}[x(T)]=d,\] \[\quad\quad\quad\{x(t),u(t)\}\ \text{ satisfies (\ref{eq:M}) and (\ref{eq:M})}.\]  
Clearly, this problem was investigated in detail in [17]. Instead of using the complicated backward stochastic differential equation method (BSDE), we solve this problem by using some properties induced from the special structure of model. Firstly, we consider the following auxiliary problem (\(\widehat{\mathcal{A}}_{\text{MV}}(\lambda)\)) by introducing the Lagrange multiplier \(\lambda\in\mathbb{R}\) for constraint \(\text{E}[x(T)]=d\):\[(\widehat{\mathcal{A}}_{\text{MV}}(\lambda)):\ \min_{\{u(t)\}\}_{|t=0} \ \ \ \text{E}[(x(T)-\lambda)^{2}]\] \[(s.t.)\ \ \{x(t),u(t)\}\ \ \text{satisfies (\ref{eq:A_MV}) and (\ref{eq:A_MV}).}\]  
Define the value function of problem \((\widehat{\mathcal{A}}_{\text{MV}}(\lambda))\) at any time \(t\in[0,T]\) as follows:  
\[V(t,x):= \ \min_{\{H(s)u(s)\geq 0\}\mid_{t=t}^{T}}\ \text{E}_{t}[(x(T)- \lambda)^{2}]\] \[(s.t.)\ \ dx(s)=\Big{(}r(s)x(s)+b(s)^{\prime}u(s)\Big{)}dt+u(s)^{ \prime}\sigma(s)dW(s),\ \ s\in[t,T].\]  
From the classical optimal control theory (see, e.g., [19; 20]), the value function \(V(t,x)\) satisfies the following Hamilton-Jacobi-Bellman (HJB) equation:  
\[V_{t}(t,x)+\inf_{u(t):H(t)u(t)\geq 0}\{\frac{1}{2}V_{xx}(t,x)u(t)^{ \prime}\sigma(t)\sigma(t)^{\prime}u(t)+V_{x}(t,x)[r(t)x+b(t)^{\prime}u(t)]\} =0, \tag{14}\]  
with the boundary condition \(V(T,x)=(x-\lambda)^{2}\). By Theorem 1 in [18], we have the following important result. **Theorem 2**. : _While \(V_{xx}(t,x)\
eq 0\), the infimum in the HJB Equation (14) can be obtained as follows:_  
\[u^{*}(t)=\frac{V_{x}(t,x)}{V_{xx}(t,x)}\hat{K}(t)\mathbf{1}_{\frac{V_{x}(t,x)} {V_{xx}(t,x)}\geq 0}+\Big{|}\frac{V_{x}(t,x)}{V_{xx}(t,x)}\Big{|}\hat{K}(t) \mathbf{1}_{\frac{V_{x}(t,x)}{V_{xx}(t,x)}<0^{\prime}}\]  
_where the \(\hat{K}(t)\) and \(\hat{K}(t)\) are optimal solutions of the following problems:_  
\[\min_{K(t):H(t)K(t)\geq 0}\ \frac{1}{2}K(t)^{\prime}\sigma(t)\sigma(t)^{ \prime}K(t)+b(t)^{\prime}K(t)\ \ \text{and}\ \ \min_{K(t):H(t)K(t)\geq 0}\ \frac{1}{2}K(t)^{\prime}\sigma(t)\sigma(t)^{ \prime}K(t)-b(t)^{\prime}K(t),\]  
_respectively._  
Proof. : As for getting the result using Theorem 1 in [18], we leave the proof for the interested readers. Before we go further, we need the following result. **Lemma 2**. : _The value function \(V(t,\cdot)\) is convex on \(\mathbb{R}\), strictly decreasing on \((-\infty,\lambda\rho(t)]\) and strictly increasing on \((\lambda\rho(t),+\infty)\), for each fixed \(t\in[0,T]\)._  
From Theorem 2 and Lemma 2, we can obtain the optimal portfolio policy of problem \((\mathcal{A}_{\text{MV}})\). Before we go further, we must introduce the following two ODEs for the two unknown functions \(\hat{F}(\cdot):\mathbb{R}\to\mathbb{R}\) and \(\hat{F}(\cdot):\mathbb{R}\to\mathbb{R}\):  
\[\quad\quad(s.t): \text{E}[x(T)]=d,\]0 \[\quad\quad(s.t): \text{E}[x(T)]=d,\]1  
with the boundary condition \(\hat{F}(T)\)=\(\hat{F}(T)=1\) and  
\[\quad\quad(s.t): \text{E}[x(T)]=d,\]2 \[\quad\quad(s.t): \text{E}[x(T)]=d,\]3  
**Theorem 3**. : _The optimal portfolio policy of problem \((\mathcal{A}_{MV})\) is characterized by_  
\[u^{*}(t)= \begin{cases}\hat{L}(t)(x(t)-\lambda^{*}\rho(t))&\text{ if }x(t)- \lambda^{*}\rho(t)\geq 0,\\ -\hat{L}(t)(x(t)-\lambda^{*}\rho(t))&\text{ if }x(t)-\lambda^{*}\rho(t)<0,\end{cases}