Stochastic dynamic programming with non-linear discounting

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

Stochastic Dynamic Programming with Non-linear Discounting

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 Jaśkiewicz et al. (Math Oper Res 38:108–121, 2013), 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

Evolution as Learning Yields Hyperbolic Discounting

Learning is modeled as an infection, which jumps from person to person. The rate of infection mimics individual discount rates and induces savings behavior on its own. It is shown that the apparent discount rate, the combination of the agents' true discount rate and the infection rate, decreases over time and approaches the agents' true discount rate. This decrease, known as hyperbolic discounting, is consistent with what is observed in psychology studies, while the limiting case, exponential discounting, is consistent with market level observations. This model closes the gap between individual and market level observations of discounting behavior without explicitly assuming the two kinds of discounting nor relying on commitment mechanisms.discounting, genetic algorithms, learning

On Variable Discounting in Dynamic Programming: Applications to Resource Extraction and Other Economic Models

This paper generalizes the classical discounted utility model introduced by Samuelson by replacing a constant discount rate with a function. The existence of recursive utilities and their constructions are based on Matkowski's extension of the Banach Contraction Principle. The derived utilities go beyond the class of recursive utilities studied in the literature and enable a discussion on subtle issues concerning time preferences in the theory of finance, economics or psychology. Moreover, our main results are applied to the theory of optimal growth with unbounded utility functions

Growth-optimal portfolios under transaction costs

This paper studies a portfolio optimization problem in a discrete-time Markovian model of a financial market, in which asset price dynamics depend on an external process of economic factors. There are transaction costs with a structure that covers, in particular, the case of fixed plus proportional costs. We prove that there exists a self-financing trading strategy maximizing the average growth rate of the portfolio wealth. We show that this strategy has a Markovian form. Our result is obtained by large deviations estimates on empirical measures of the price process and by a generalization of the vanishing discount method to discontinuous transition operators.Comment: 32 page

On Variable Discounting in Dynamic Programming: Applications to Resource Extraction and Other Economic Models

This paper generalizes the classical discounted utility model introduced by Samuelson by replacing a constant discount rate with a function. The existence of recursive utilities and their constructions are based on Matkowski's extension of the Banach Contraction Principle. The derived utilities go beyond the class of recursive utilities studied in the literature and enable a discussion on subtle issues concerning time preferences in the theory of finance, economics or psychology. Moreover, our main results are applied to the theory of optimal growth with unbounded utility functions

Coherent Measures of Risk from a General Equilibrium Perspective

Coherent measures of risk defined by the axioms of monotonicity, subadditivity, positive homogeneity, and translation invariance are recent tools in risk management to assess the amount of risk agents are exposed to. If they also satisfy law invariance and comonotonic additivity, then we get a subclass of them: spectral measures of risk. Expected shortfall is a well-known spectral measure of risk is. We investigate the above mentioned six axioms using tools from general equi- librium (GE) theory. Coherent and spectral measures of risk are compared to the natural measure of risk derived from an exchange economy model, that we call GE measure of risk. We prove that GE measures of risk are coherent measures of risk. We also show that spectral measures of risk can be represented by GE measures of risk only under stringent conditions, since spectral measures of risk do not take the regulated entity's relation to the market portfolio into account. To give more insights, we characterize the set of GE measures of risk.Coherent Measures of Risk, General Equilibrium Theory, Exchange Economies, Asset Pricing