The Annals of Applied Probability

Dynamic approaches for some time-inconsistent optimization problems

Chandrasekhar Karnam, Jin Ma, and Jianfeng Zhang

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 6 (2017), 3435-3477.

Dates
Received: April 2016
Revised: January 2017
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1513328705

Digital Object Identifier
doi:10.1214/17-AAP1284

Mathematical Reviews number (MathSciNet)
MR3737929

Zentralblatt MATH identifier
1385.49019

Subjects
Primary: 49L20: Dynamic programming method 60H10: Stochastic ordinary differential equations [See also 34F05] 91C99: None of the above, but in this section 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25]

Keywords
Time inconsistency dynamic programming principle stochastic maximum principle comparison principle duality dynamic utility master equation path derivative

Citation

Karnam, Chandrasekhar; Ma, Jin; Zhang, Jianfeng. Dynamic approaches for some time-inconsistent optimization problems. Ann. Appl. Probab. 27 (2017), no. 6, 3435--3477. doi:10.1214/17-AAP1284. https://projecteuclid.org/euclid.aoap/1513328705


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