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We study Merton's expected utility maximization problem in an incompletemarket, characterized by a factor process in addition to the stock priceprocess, where all the model primitives are unknown. We take the reinforcementlearning (RL) approach to learn optimal portfolio policies directly byexploring the unknown market, without attempting to estimate the modelparameters. Based on the entropy-regularization framework for generalcontinuous-time RL formulated in Wang et al. (2020), we propose a recursiveweighting scheme on exploration that endogenously discounts the currentexploration reward by the past accumulative amount of exploration. Such arecursive regularization restores the optimality of Gaussian exploration.However, contrary to the existing results, the optimal Gaussian policy turnsout to be biased in general, due to the interwinding needs for hedging and forexploration. We present an asymptotic analysis of the resulting errors to showhow the level of exploration affects the learned policies. Furthermore, weestablish a policy improvement theorem and design several RL algorithms tolearn Merton's optimal strategies. At last, we carry out both simulation andempirical studies with a stochastic volatility environment to demonstrate theefficiency and robustness of the RL algorithms in comparison to theconventional plug-in method.

Learning Merton's Strategies in an Incomplete Market: Recursive Entropy Regularization and Biased Gaussian Exploration