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We introduce a novel class of semilinear nonlocal backward stochastic partialdifferential equations (BSPDE) on half-spaces driven by an infinite-dimensionalc\`adl\`ag martingale. The equations exhibit a degeneracy and have no explicitcondition at the boundary of the half-space. To treat the existence anduniqueness of such BSPDEs we establish a generalisation of It\^o's formula forinfinite-dimensional c\`adl\`ag semimartingales, which addresses the occurrenceof boundary terms. Next, we employ these BSPDEs to study the McKean--Vlasovcontrol problem with killing and common noise proposed in [HJ23]. The particlesin this control model live on the real line and are killed at a positiveintensity whenever they are in the negative half-line. Accordingly, theinteraction between particles occurs through the subprobability distribution ofthe living particles. We establish the existence of an optimal semiclosed-loopcontrol that only depends on the particles' location and not their cumulativeintensity. This problem cannot be addressed through classical mimickingarguments, because the particles' subprobability distribution cannot bereconstructed from their location alone. Instead, we represent optimal controlsin terms of the solutions to BSPDEs of the above type and show those solutionsdo not depend on the intensity variable.

Semilinear BSPDEs and Applications to McKean-Vlasov Control with Killing