We consider the problem of stopping a diffusion process with a payoff functional that renders the problem time‐inconsistent. We study stopping decisions of naïve 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 naïve 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.”

中文翻译：

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幼稚和不负责任的复杂智能体的一般停止行为，应用于概率失真

我们考虑了使用支付功能停止扩散过程的问题，该支付功能使该问题在时间上不一致。我们研究了及时进行连续优化的幼稚代理商的停止决策，以及预期但无法控制其未来自我行为的成熟代理商的均衡策略。当状态过程为一维且收益函数满足某些规则性条件时，我们证明可以将任何均衡作为算子的不动点获得。该运算符表示战略推理，它考虑了未来自我的行为。然后，我们将一般结果应用于当代理商扭曲概率且扩散过程为几何布朗运动的情况。该问题本质上是时间不一致的，因为同一事件的失真程度会随着时间而变化。我们展示了战略推理如何将幼稚的主体变成复杂的主体。此外，我们针对问题的各种参数规格推导了两种类型的代理的停止策略，从而说明了超出极端行为（例如“永不停止”或“永不启动”）的丰富行为。