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December 2017 Dynamic approaches for some time-inconsistent optimization problems
Chandrasekhar Karnam, Jin Ma, Jianfeng Zhang
Ann. Appl. Probab. 27(6): 3435-3477 (December 2017). DOI: 10.1214/17-AAP1284


In this paper, we investigate possible approaches to study general time-inconsistent optimization problems without assuming the existence of optimal strategy. This leads immediately to the need to refine the concept of time consistency as well as any method that is based on Pontryagin’s maximum principle. The fundamental obstacle is the dilemma of having to invoke the Dynamic Programming Principle (DPP) in a time-inconsistent setting, which is contradictory in nature. The main contribution of this work is the introduction of the idea of the “dynamic utility” under which the original time-inconsistent problem (under the fixed utility) becomes a time-consistent one. As a benchmark model, we shall consider a stochastic controlled problem with multidimensional backward SDE dynamics, which covers many existing time-inconsistent problems in the literature as special cases; and we argue that the time inconsistency is essentially equivalent to the lack of comparison principle. We shall propose three approaches aiming at reviving the DPP in this setting: the duality approach, the dynamic utility approach and the master equation approach. Unlike the game approach in many existing works in continuous time models, all our approaches produce the same value as the original static problem.


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Chandrasekhar Karnam. Jin Ma. Jianfeng Zhang. "Dynamic approaches for some time-inconsistent optimization problems." Ann. Appl. Probab. 27 (6) 3435 - 3477, December 2017.


Received: 1 April 2016; Revised: 1 January 2017; Published: December 2017
First available in Project Euclid: 15 December 2017

zbMATH: 1385.49019
MathSciNet: MR3737929
Digital Object Identifier: 10.1214/17-AAP1284

Primary: 35R15, 49L20, 60H10, 91C99, 91G80

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.27 • No. 6 • December 2017
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