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arXiv · 2011.02239

Stochastic dynamic programming with non-linear discounting

Nicole Bäuerle·Anna Jaśkiewicz·Andrzej S. Nowak
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Abstract

In this paper, we study a Markov decision process with a non-linear discount function and with a Borel state space. We define a recursive discounted utility, which resembles non-additive utility functions considered in a number of models in economics. Non-additivity here follows from non-linearity of the discount function. Our study is complementary to the work of Ja\'skiewicz, Matkowski and Nowak (Math. Oper. Res. 38 (2013), 108-121), where also non-linear discounting is used in the stochastic setting, but the expectation of utilities aggregated on the space of all histories of the process is applied leading to a non-stationary dynamic programming model. Our aim is to prove that in the recursive discounted utility case the Bellman equation has a solution and there exists an optimal stationary policy for the problem in the infinite time horizon. Our approach includes two cases: $(a)$ when the one-stage utility is bounded on both sides by a weight function multiplied by some positive and negative constants, and $(b)$ when the one-stage utility is unbounded from below.

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Nicole Bäuerle, Anna Jaśkiewicz, Andrzej S. Nowak. 2020-11-04. Stochastic dynamic programming with non-linear discounting. https://doi.org/10.1007/s00245-020-09731-x

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