  We prove that the weak version of the SPDE problem \begin{align*} dV_{t}(x) &
= [-\mu V_{t}'(x) + \frac{1}{2} (\sigma_{M}^{2} + \sigma_{I}^{2})V_{t}"(x)]dt -
\sigma_{M} V_{t}'(x)dW^{M}_{t}, \quad x > 0, \\ V_{t}(0) &= 0 \end{align*} with
a specified bounded initial density, $V_{0}$, and $W$ a standard Brownian
motion, has a unique solution in the class of finite-measure valued processes.
The solution has a smooth density process which has a probabilistic
representation and shows degeneracy near the absorbing boundary. In the
language of weighted Sobolev spaces, we describe the precise order of
integrability of the density and its derivatives near the origin, and we relate
this behaviour to a two-dimensional Brownian motion in a wedge whose angle is a
function of the ratio $\sigma_{M}/\sigma_{I}$. Our results are sharp: we
demonstrate that better regularity is unattainable.
