何时应停止最激动人心的游戏？赢鞅的序贯推断
When should one stop the most exciting game? Sequential Inference for win-martingales
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中文总结 AI 辅助
本文针对赢鞅的最优停止问题，构建了通用决策理论框架，推导了其解的完整自由边界刻画等结果，可用于预测市场等场景中决策时机的确定。
中文摘要 AI 辅助
预测市场已成为聚合关于二元未来事件信念的重要方式，其价格过程常被解读为不断演化的获胜概率，即“赢鞅”。受这一视角及Aldous提出的“最激动人心的游戏”相关近期研究的启发，本文研究决策者应在何时停止观测赢鞅并对结果作出决策。具体而言，本文允许真实结果在固定有限时间范围内揭晓，如同体育赛事或选举场景。在一般终端损失与运行成本下，本文将贝叶斯风险简化为针对获胜概率过程的最优停止问题。当赢鞅为扩散过程且其波动率可分解为确定性时间因子与状态依赖因子时，确定性时间变换可将该问题转化为具有一般非齐次运行成本的时间齐次扩散问题。在明确的结构假设下，本文在有限与无限时间范围（无贴现）内得到了停止问题解的完整自由边界刻画；证明了值函数的光滑拟合与C¹正则性、时间范围前最优停止边界的C¹正则性，并推导了唯一刻画边界的非线性积分方程。综上，这些结果为一类广泛的后验动态提供了通用决策理论框架与解理论，其中包括Aldous鞅、Bass鞅及二元序贯推断鞅作为特例。本文的分析不要求运行成本具有时间单调性，因此可容纳高度非单调的停止边界，特别地，本文给出了一个示例，其中最优边界在接近日历时间范围时无极限。
英文摘要
Prediction markets have become a prominent way of aggregating beliefs about binary future events, and their price processes are often interpreted as evolving win probabilities, or ``win-martingales.'' Motivated by this perspective and recent work on Aldous' ``most exciting game,'' we study when a decision maker should stop observing a win-martingale and make a decision about the outcome. In particular, we allow the true outcome to be revealed at a fixed finite horizon, as in a sports game or election. Under a general terminal loss and running cost, we reduce the Bayes risk to an optimal stopping problem for the win probability process. When the win-martingale is a diffusion and its volatility separates into a deterministic time factor and a state-dependent factor, a deterministic time change transforms the problem into one for a time-homogeneous diffusion with a generally time-inhomogeneous running cost. Under explicit structural assumptions, we obtain a complete free-boundary characterization of the solution to the stopping problem in both finite and infinite horizons without discounting. We prove smooth-fit and $C^1$ regularity of the value function, $C^1$ regularity of the optimal stopping boundaries before the horizon, and derive a nonlinear integral equation that characterizes the boundaries uniquely. Taken together, these results yield a common decision-theoretic framework and solution theory for a broad class of posterior dynamics that includes the Aldous, Bass, and binary sequential-inference martingales as special cases. Our analysis requires no temporal monotonicity of the running cost and therefore accommodates highly nonmonotone stopping boundaries. In particular, we exhibit an example in which the optimal boundary has no limit as the calendar-time horizon is approached.
