Introduction: The integration of ESG criteria into portfolio management has grown substantially, yet the theoretical literature remains largely static: ESG scores are treated as deterministic or single-period random variables. In practice, scores evolve as firms change their environmental practices and governance structures, exposing multi-year investors to ESG score risk. This paper addresses this gap by extending the ESG portfolio problem to continuous time.

Methods. We derive the optimal dynamic investment strategy for a CARA investor with warm-glow preferences over portfolio ESG quality, where scores follow a multivariate Ornstein–Uhlenbeck process correlated with asset returns. The solution to the associated Hamilton–Jacobi–Bellman equation can be characterized explicitly. We further extend the model to incorporate ambiguity aversion about ESG dynamics using a Hansen–Sargent robust control formulation.

Results. The optimal portfolio decomposes into four components, two of which correspond to intertemporal hedging demands against ESG score risk. Closed-form solutions are derived in the scalar case, with comparative statics showing that the hedging demand grows quadratically in ESG preference intensity, depends critically on the return–ESG correlation, and saturates with the investment horizon. We extend the model to incorporate ambiguity aversion about ESG dynamics. The model nests the static frameworks of Pástor et al. (2021), Pedersen et al. (2021), and Avramov et al. (2022) as limiting cases.

Conclusions. Dynamic ESG investing generates hedging demands absent from static frameworks, whose magnitude depends on horizon, persistence, and return–ESG correlation. Ambiguity surroundng ESG dynamics attenuates but does not eliminate these demands, providing a tractable bridge between static and fully dynamic approaches.