  This paper introduces a novel stochastic control framework to enhance the
capabilities of automated investment managers, or robo-advisors, by accurately
inferring clients' investment preferences from past activities. Our approach
leverages a continuous-time model that incorporates utility functions and a
generic discounting scheme of a time-varying rate, tailored to each client's
risk tolerance, valuation of daily consumption, and significant life goals. We
address the resulting time inconsistency issue through state augmentation and
the establishment of the dynamic programming principle and the verification
theorem. Additionally, we provide sufficient conditions for the identifiability
of client investment preferences. To complement our theoretical developments,
we propose a learning algorithm based on maximum likelihood estimation within a
discrete-time Markov Decision Process framework, augmented with entropy
regularization. We prove that the log-likelihood function is locally concave,
facilitating the fast convergence of our proposed algorithm. Practical
effectiveness and efficiency are showcased through two numerical examples,
including Merton's problem and an investment problem with unhedgeable risks.
  Our proposed framework not only advances financial technology by improving
personalized investment advice but also contributes broadly to other fields
such as healthcare, economics, and artificial intelligence, where understanding
individual preferences is crucial.
