Solution
= Solution
For
$$
F(t,S,v)=S^{1/2}e^{A(t)v+B(t)},
$$
one has
$$
\frac{F_S}{F}=\frac1{2S},
\quad
\frac{F_{SS}}F=-\frac1{4S^2},
\quad
\frac{F_v}F=A,
\quad
\frac{F_{Sv}}F=\frac{A}{2S},
\quad
\frac{F_{vv}}F=A^2.
$$
Substitution into the PDE and collection of the coefficient of $v$ give the <Riccati differential equation>
$$
\boxed{
A'(t)-bA(t)-\frac18
+\frac{c\rho}{2}A(t)
+\frac{c^2}{2}A(t)^2=0,
\qquad A(T)=0.}
$$
The terminal condition also requires $B(T)=0$.