Entropy-Regularized Entropic Recursive Utility in Complete Black-Scholes Markets

This paper studies continuous-time consumption-investment decisions in a complete multi-asset Black-Scholes market under \emph{entropic recursive utility} represented by a quadratic BSDE. To incorporate robustness and exploration in the control layer, portfolio choice is formulated as an \emph{entropy-regularized relaxed control} relative to a Gaussian reference prior. By applying the Donsker-Varadhan variational formula, the resulting dynamic programming equation admits a \emph{soft Hamiltonian} of log-sum-exp type, and the optimal relaxed portfolio is characterized by an explicit \emph{Gibbs tilt} of the prior. Under log utility, we derive a closed-form Markovian ansatz for the value function and reduce the associated soft HJB equation to a system of scalar ODEs. We then provide a verification theorem under explicit well-posedness and admissibility conditions tailored to quadratic BSDE preferences, including BMO-martingale requirements for the candidate martingale integrand. Finally, we outline and numerically validate an extension to Epstein-Zin recursive preferences via a Deep BSDE solver that preserves the entropy-regularized structure beyond the log closed-form benchmark, together with stability diagnostics and risk-return sensitivity modes.