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An Inequality for Variances of the Discounted Rewards

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Eugene A. Feinberg  and
Jun Fei
Eugene A. Feinberg*
Affiliation:
Stony Brook University
Jun Fei*
Affiliation:
Stony Brook University
*
∗Postal address: Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA.
∗Postal address: Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA.
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Abstract

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We consider the following two definitions of discounting: (i) multiplicative coefficient in front of the rewards, and (ii) probability that the process has not been stopped if the stopping time has an exponential distribution independent of the process. It is well known that the expected total discounted rewards corresponding to these definitions are the same. In this note we show that, the variance of the total discounted rewards is smaller for the first definition than for the second definition.

Keywords

Total discounted reward; variancestopping time

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 90C40: Markov and semi-Markov decision processes
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1209 - 1212
Copyright
Copyright © Applied Probability Trust 2009 

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