  We study a finite system of diffusions on the half-line, absorbed when they
hit zero, with a correlation effect that is controlled by the proportion of the
processes that have been absorbed. As the number of processes in the system
becomes large, the empirical measure of the population converges to the
solution of a non-linear stochastic heat equation with Dirichlet boundary
condition. The diffusion coefficients are allowed to have finitely many
discontinuities (piecewise Lipschitz) and we prove pathwise uniqueness of
solutions to the limiting stochastic PDE. As a corollary we obtain a
representation of the limit as the unique solution to a stochastic
McKean--Vlasov problem. Our techniques involve energy estimation in the dual of
the first Sobolev space, which connects the regularity of solutions to their
boundary behaviour, and tightness calculations in the Skorokhod M1 topology
defined for distribution-valued processes, which exploits the monotonicity of
the loss process $L$. The motivation for this model comes from the analysis of
large portfolio credit problems in finance.
