EXERCISE 23.1 This exercise may be treated as a version of the Robinson Crusoe problem of Paragraph $19 \mathrm{~A}$ in a continuous-time setting. The rate of growth of the capital stock is determined by a "random shock" process $Y$ defined as the solution to the stochastic differential equation
$$
d Y_t=\left(a Y_t-b\right) d t+k \bar{Y}_t d B_t, \quad Y_0=y \in R, t \geq 0
$$
where $a, b$, and $k$ are strictly positive scalars with $b \geq 0$, and $B$ is a Standard Brownian Motion that is a martingale with respect to the given filtered probability space. A capital stock process $K$ is defined by
$$
K_t=\kappa+\int_0^t\left(K_s h Y_s-c_s\right) d s+\int_0^t K_s \epsilon \sqrt{Y_s} d B_s, \quad t \geq 0,
$$
where $h, \kappa$, and $\epsilon$ are strictly positive scalars and $c$ is an element of the space $\mathcal{C}$ of positive predictable consumption processes $c=\left\{c_t: t \geq 0\right\}$ satisfying $\int_0^T c_t d t<\infty$ almost surely for all $T \geq 0$. A given agent has the control problem
$$
V(\kappa, y, 0)=\sup _{c \in \mathcal{C}} E\left[\int_0^T e^{-\rho t} \log \left(c_t\right) d t\right],
$$
subject to $K_t \geq 0$ for all $t$ in $[0, T]$, where $T$ is a scalar time horizon.
(A) Show that the unique optimal consumption control is $\left\{\rho K_t /(1-\right.$ $\left.\left.e^{-p(T-t)}\right): t \geq 0\right\}$, and that the value function of is of the form
$$
V(\kappa, y, t)=A_1(t) \log (\kappa)+A_2(t) y+A_3(t), \quad(\kappa, y, t) \in R_{+} \times R \times[0, T),
$$
where $A_1, A_2$, and $A_3$ are (deterministic) real-valued functions of time. State the function $A_1$ and a differential equation for $A_2$ and $A_3$.
(B) Explicitly state the value function and the optimal consumption control for the infinite horizon case.