I would like to contribute material on continuous time optimization, but there is one thing that is not clear to me.

Consider a routine continuous time optimization problem:

\max \int_{t=0}^{T} e^{-\rho t}u(c_{t})dt \text{ s.t. }

\dot{a}_{t} = y + ra_{t} - c_{t},

a_{0} \text{ given, } a_{T}=0.

Assume y & r are constants and u(c)=\frac{c^{1-\gamma}}{1-\gamma}.

Write the value function V(t,a_{t})

**Optimal control approach**:

H\equiv u(c_{t}) + \lambda (y + ra_{t} - c_{t})

Boils down to a BVP (two ODEs w/ unkown c_{t}, a_{t}):

\left[\begin{array}{l}
\dot{c}=\left(\frac{r-\rho}{\gamma} \right) c
\\
\dot{a}_{t} = y + ra_{t} - c_{t}
\\
a(0)=a_0
\\
a_{T}=0
\end{array} \right]

**HJB approach**:

\rho V(t,a_{t}) = \max_{c} \left\{u(c_{t}) + V_{a}(t,a_{t})\times \left(y + ra_{t} - c_{t} \right) + V_{t}(t,a_{t}) \right\}

FOC: c(t,a_{t})=u'^{-1}(V_{a}(t,a_{t})) = (V_{a}(t,a_{t}))^{-\frac{1}{\gamma}}

This boils down to a PDE w/ unknown function V(t,a_{t}):

\left[\begin{array}{l}
\rho V(t,a_{t}) = u(c_{t}) + V_{a}(t,a_{t})\times \left(y + ra_{t} - c_{t} \right) + V_{t}(t,a_{t})
\\
c(t,a_{t}) = (V_{a}(t,a_{t}))^{-\frac{1}{\gamma}}
\\
a(0)=a_0
\\
a_{T}=0
\end{array} \right]

**Issue**: we have boundary conditions for a_{t} (a(0)=a_0, a_{T}=0) but not for V(t,a_{t}).

To solve this PDE we need two boundary conditions for V(t,a_{t}).

**Question**: How do we get the two boundary conditions?

I think one of them is: V(T,a_{T}) = \psi(a_{T}) = 0 (no bequest here)

Or maybe something like:

\left[\begin{array}{l}
\rho V(t,a_{t}) = u(c_{t}) + V_{a}(t,a_{t})\times \left(y + ra_{t} - c_{t} \right) + V_{t}(t,a_{t})
\\
c(t,a_{t}) = (V_{a}(t,a_{t}))^{-\frac{1}{\gamma}}
\\
V(T,a_{T}) = 0
\\
V_{a}(T,a_{T}) = 0
\end{array} \right]