Robust optimal reinsurance and ESG investment under rating-innovation ambiguity and ESG rating uncertainty

ESG ratings are noisy measurements and may also be exposed to systematic credibility concerns. We study an ambiguity-averse insurer that chooses proportional reinsurance and green and brown stock positions while learning a hidden green-quality factor from stock returns and an ESG rating signal. Signal volatility represents ordinary rating uncertainty. A worst-case distortion of the rating innovation represents ambiguity about the observation model, whereas a separate out-of-model drift gtrue is used only in stress tests for systematic rating bias. This distinction prevents robust-control ambiguity from being interpreted as observed greenwashing itself. We derive the reference Kalman-Bucy filter and the separated robust control problem. Shared stock-return shocks produce a mixed wealth-belief derivative and a filtering-hedging demand. Under CARA utility, the value function is exponential-quadratic, stock feedback follows a Riccati-type system, and retention is projected onto q∈[0,1]. The verification result is conditional on explicit admissibility and integrability conditions. Primary numerical experiments with illustrative parameters identify the model mechanisms but do not constitute empirical calibration. Independent numerical checks add Monte Carlo intervals, static and insurance-only benchmarks, original-scale plots, and a 40-design parameter-uncertainty analysis. The learning-strategy intervals overlap substantially, and rating-innovation ambiguity effects are small in CE and warning probability. Tail-risk changes are more directionally consistent across the tested parameter region, but their magnitudes remain small. The numerical evidence therefore supports mechanism existence and conditional downside sensitivity, not universal or quantitatively calibrated economic relevance.