Causal Price-of-Risk Mandates under Overlapping Information

Abstract

We study whether causal risk mandates constructed from overlapping information blocks can be implemented by one self-financing portfolio that is optimal under pooled information. In an incomplete continuous Brownian market, signal-responsive exposures are projected onto traded Brownian directions and embedded as closed subspaces of a predictable Hilbert space. Each causal mandate is the attainable projection of an identified, baseline-centered intervention surface and is therefore fixed upstream of the allocator. A first exact decomposition separates non-traded structural response, incompatibility across local mandates, and distortion of the common traded book. A fusion operator corrects duplication of common risk directions. The main theorem shows that exact causal decentralization holds if and only if the pooled enlargement preserves reference martingales, the pooled optimum has no component outside the additive desk span, the local causal shadows are compatible with one common book, and that book equals the additive pooled optimum. Under immersion, log-growth regret separates into pooled-interaction loss and common-book distortion, while raw causal implementation error also contains non-attainability and incompatibility. No common benchmark can reduce incompatibility. We derive the unique allocation under a quadratic causal-mandate penalty and exact comparative statics in a two-desk model with shared, desk-specific, and pooled-interaction risk. Additional results cover joint interventions, representation stability, admissible coalitions, and a covariance-weighted continuous-semimartingale formulation. The conclusions are partial-equilibrium statements for continuous markets.