## The Annals of Applied Probability

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

### Dynamic approaches for some time-inconsistent optimization problems

Chandrasekhar Karnam, Jin Ma, and Jianfeng Zhang

#### 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