论文标题
我，我和我：非马克维亚时间不一致的随机控制的一般理论，用于复杂的代理
Me, myself and I: a general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents
论文作者
论文摘要
我们开发了一种理论，用于固有时间不一致的连续时间非马克维亚随机控制问题。他们的独特特征是古典贝尔曼最佳原则不再成立。我们的公式是在受控的非马克维亚前进随机微分方程和一般客观功能设置的框架内施放的。我们采用游戏理论方法来研究此类问题，这意味着我们寻求子游戏完美的NASH平衡点。作为这项工作的第一个新颖性，我们介绍并激励了平衡的定义的改进，这使我们能够以与经典理论相同的精神建立一个直接而严格的扩展动态编程原理的证据。反过来，这使我们能够引入一个由无限的后向随机微分方程组成的系统，类似于经典的HJB方程。我们证明，该系统是基本的，因为它的体积良好既是必要的，也足以表征价值函数和平衡。作为最后一步，我们提供了存在和独特性结果。还提供了一些结果的例子和扩展。
We develop a theory for continuous-time non-Markovian stochastic control 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 forward stochastic differential equation, and a general objective functional setting. We adopt a game-theoretic approach to study such problems, meaning that we seek for sub-game perfect Nash equilibrium points. As a first novelty of this work, we introduce and motivate a refinement of the definition of equilibrium 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.
