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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*} witha specified bounded initial density, $V_{0}$, and $W$ a standard Brownianmotion, has a unique solution in the class of finite-measure valued processes.The solution has a smooth density process which has a probabilisticrepresentation and shows degeneracy near the absorbing boundary. In thelanguage of weighted Sobolev spaces, we describe the precise order ofintegrability of the density and its derivatives near the origin, and we relatethis behaviour to a two-dimensional Brownian motion in a wedge whose angle is afunction of the ratio $\sigma_{M}/\sigma_{I}$. Our results are sharp: wedemonstrate that better regularity is unattainable.

Sharp regularity near an absorbing boundary for solutions to second order SPDEs in a half-line with constant coefficients