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Most conventional Reinforcement Learning (RL) algorithms aim to optimizedecision-making rules in terms of the expected returns. However, especially forrisk management purposes, other risk-sensitive criteria such as thevalue-at-risk or the expected shortfall are sometimes preferred in realapplications. Here, we describe a parametric method for estimating density ofthe returns, which allows us to handle various criteria in a unified manner. Wefirst extend the Bellman equation for the conditional expected return to covera conditional probability density of the returns. Then we derive an extensionof the TD-learning algorithm for estimating the return densities in an unknownenvironment. As test instances, several parametric density estimationalgorithms are presented for the Gaussian, Laplace, and skewed Laplacedistributions. We show that these algorithms lead to risk-sensitive as well asrobust RL paradigms through numerical experiments.

Parametric Return Density Estimation for Reinforcement Learning