Anybody have an ansatz for the following PDE on $[0,T) \times \mathbb{R}$:
$$ V_t + \frac{1}{2\lambda} V_x^2 + \frac{\sigma^2}{2} V_{xx} = 0, \quad V(T,x) = g(x) = -(x-b)^2. $$
$V(t,x)$ is the value function for the stochastic control problem
$$ V(t,x) = \max_{(a_t)} E^{t,x} [\int_t^T -\frac{\lambda}{2} a_s^2 ds + g(X^{t,x}_T)] $$ where $$ dX_t = a_t dt + \sigma dW_t, \quad \mbox{$W_t$ standard Brownian motion}. $$
The corresponding HJB equation is
$$ V_t + \sup_a \{ a V_x + \frac{\sigma^2}{2} V_{xx} - \frac{\lambda}{2} a^2\} =0, $$
which gives the PDE above.
You mean $V_{xx}$ instead of $V_xx$ in the pde. Since the final value is quadratic in $x$, and since such functions are closed under the operations in the pde, that suggests trying $$V(x,t) = a(t)+b(t)x+c(t)x^2,$$ In fact that works because it produces three easy odes.