RENEWAL THEORY WITH EXPONENTIAL AND HYPERBOLIC DISCOUNTING

To determine optimal investment and maintenance decisions, the total costs should be minimized over the whole life of a system or structure. In minimizing life-cycle costs, it is important to account for the time value of money by discounting and to consider the uncertainties involved. This article presents new results in renewal theory with costs that can be discounted according to any discount function that is nonincreasing and monotonic over time (such as exponential, hyperbolic, generalized hyperbolic, and no discounting). The main results include expressions for the first and second moment of the discounted costs over a bounded and unbounded time horizon as well as asymptotic expansions for nondiscounted costs.

[1]  C. M. Bradshaw,et al.  Quantification of Steady-State Operant Behavior , 1982 .

[2]  P. Samuelson.  A Note on Measurement of Utility , 1937 .

[3]  R. Pollak,et al.  SECOND-BEST NATIONAL SAVING AND GAME-EQUILIBRIUM GROWTH , 1980 .

[4]  Xiong Shuang-ping.  Moments of Compound Renewal Sums with Discounted Claims , 2005 .

[5]  Walter L. Smith.  Renewal Theory and its Ramifications , 1958 .

[6]  Walter L. Smith.  II.—Asymptotic Renewal Theorems , 1953, Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences.

[7]  Present Values with Renewals , 1992 .

[8]  Rüdiger Rackwitz.  Optimizing systematically renewed structures , 2001, Reliab. Eng. Syst. Saf..

[9]  Richard C. Morey,et al.  Some Stochastic Properties of a Compound-Renewal Damage Model , 1966, Oper. Res..

[10]  Henk C. Tijms,et al.  A First Course in Stochastic Models: Tijms/Stochastic Models , 2003 .

[11]  P. Iseger.  NUMERICAL TRANSFORM INVERSION USING GAUSSIAN QUADRATURE , 2005, Probability in the Engineering and Informational Sciences.

[12]  J. E. Mazur.  An adjusting procedure for studying delayed reinforcement. , 1987 .

[13]  Jan M. van Noortwijk,et al.  Explicit formulas for the variance of discounted life-cycle cost , 2003, Reliab. Eng. Syst. Saf..

[14]  Rüdiger Rackwitz,et al.  Optimization — the basis of code-making and reliability verification , 2000 .

[15]  Simon French,et al.  Valuing the Future: a MADA example involving nuclear waste storage , 1998 .

[16]  G. Loewenstein,et al.  Anomalies in Intertemporal Choice: Evidence and an Interpretation , 1992 .

[17]  Walter L. Smith.  On necessary and sufficient conditions for the convergence of the renewal density , 1962 .

[18]  R. Rackwitz,et al.  Socio-economically sustainable civil engineering infrastructures by optimization , 2005 .

[19]  Ronald W. Wolff,et al.  Stochastic Modeling and the Theory of Queues , 1989 .

[20]  Casey A. Volino,et al.  A First Course in Stochastic Models , 2005, Technometrics.

[21]  G. Loewenstein,et al.  Time Discounting and Time Preference: A Critical Review , 2002 .

[22]  M. Weitzman.  Why the Far-Distant Future Should Be Discounted at Its Lowest Possible Rate , 1998 .

[23]  J. Shoven,et al.  Gamma Discounting , 2001 .