ME, MYSELF AND I: A GENERAL THEORY OF NON-MARKOVIAN TIME-INCONSISTENT STOCHASTIC CONTROL FOR SOPHISTICATED AGENTS

被引:18
作者
Hernandez, Camilo [1 ]
Possamai, Dylan [2 ]
机构
[1] Imperial Coll London, Math Dept, London, England
[2] Swiss Fed Inst Technol, Math Dept, Zurich, Switzerland
关键词
Time inconsistency; consistent planning; nonexponential discounting; mean-variance; backward stochastic differential equations; EQUILIBRIUM;
D O I
10.1214/22-AAP1845
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
We develop a theory for continuous-time non-Markovian stochastic con-trol problems which are inherently time-inconsistent. Their distinguishing feature is that the classical Bellman optimality principle no longer holds. Our formulation is cast within the framework of a controlled non-Markovian for-ward stochastic differential equation, and a general objective functional set-ting. We adopt a game-theoretic approach to study such problems, meaning that we seek for subgame perfect Nash equilibrium points. As a first novelty of this work, we introduce and motivate a refinement of the definition of equi-librium that allows us to establish a direct and rigorous proof of an extended dynamic programming principle, in the same spirit as in the classical theory. This in turn allows us to introduce a system consisting of an infinite family of backward stochastic differential equations analogous to the classical HJB equation. We prove that this system is fundamental, in the sense that its well-posedness is both necessary and sufficient to characterise the value function and equilibria. As a final step, we provide an existence and uniqueness result. Some examples and extensions of our results are also presented.
引用
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页码:1196 / 1258
页数:63
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