摘要：We consider the problem of stopping a diffusion process with a payoff functional that renders the problem time-inconsistent. We study stopping decisions  of naive agents who reoptimize continuously in time, as well as equilibrium strategies of sophisticated agents who anticipate but lack control over their future selves' behaviors. When the state process is one dimensional and the payoff functional satisfies some regularity conditions, we prove that any equilibrium can be obtained as a fixed point of an operator. This operator represents strategic reasoning that takes the future selves' behaviors into account. We then apply the general results to the case when the agents distort probability and the diffusion process is a geometric Brownian motion. The problem is inherently time-inconsistent as the level of distortion of a same event changes over time. We show how the strategic reasoning may turn a naive agent into a sophisticated one. Moreover, we derive stopping strategies of the two types of agent for various parameter specifications of the problem, illustrating rich behaviors beyond the extreme ones such as ``never-stopping" or ``never-starting".

简介：Xunyu Zhou is the Liu Family Professor of Financial Engineering at Columbia University in New York. His research focuses on quantitative behavioral finance models that incorporate human emotions and psychology into financial decision makings, and on intelligent wealth management solutions using optimal control and machine learning techniques. He directs the FDT Center for Intelligent Asset Management, a research center funded by a FinTech company. He has addressed the 2010 International Congress of Mathematicians, and has been awarded the Wolfson Research Award from The Royal Society (UK), the Outstanding Paper Prize from SIAM, the Humboldt Distinguished Lecturer and the Alexander von Humboldt Research Fellowship. He is both an IEEE Fellow and a SIAM Fellow.

Zhou received his Ph.D. in Operations Research and Control Theory from Fudan University in 1989. He was the Nomura Professor of Mathematical Finance and the Director of Nomura Center for Mathematical Finance at University of Oxford during 2007-2016 before joining Columbia.

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