AI 中文总结
本文研究有限离散时间市场中受限信息下的可选投影完备性，证明固定投影下索赔可达的充要条件，并揭示投影完备性与测度唯一性关系的非对称性，且通过二项模型延迟示例展示其与原始复制的区别。
AI 中文摘要
在有限离散时间市场中，交易决策可能相对于一个不适应资产价格的过滤是可预测的。第一基本定理随后通过测度刻画了无套利的存在性，在这些测度下，折现价格的可选投影是一个鞅。我们考察相应的完备性问题。对于固定的投影，当且仅当该固定过程的所有等价鞅测度在索赔σ-域上一致时，每个关于其终端价格历史可测的索赔都是可达的。我们给出有限维证明，保留交易过滤与索赔σ-域之间的区别。如果该投影是所有原始价格的可选鞅测度所共有的，则投影完备性蕴含这些测度限制的唯一性，但反之即使在具有唯一可选鞅测度的三状态模型中也失败。对于二项模型在其乘积鞅测度下，延迟k期产生一个具有有效期限(T-k)^+的完备投影市场。一个显式的复制构造和有限例子将这种完备性与原始价格下的复制区分开来。
英文摘要
In a finite discrete-time market, trading decisions may be predictable with respect to a filtration that does not adapt asset prices. The first fundamental theorem then characterizes absence of arbitrage by measures under which the optional projection of discounted prices is a martingale. We examine the corresponding completeness question. For a fixed projection, every claim measurable with respect to its terminal price history is attainable precisely when all equivalent martingale measures of that fixed process agree on the claim sigma-field. We give the finite-dimensional proof, retaining the distinction between the trading filtration and the claim sigma-field. If the projection is common to all optional martingale measures of the original prices, projected completeness implies uniqueness of their restrictions, but the converse fails even in a three-state model with a unique optional martingale measure. For the binomial model under its product martingale measure, a delay of $k$ periods yields a complete projected market with effective horizon $(T-k)^+$. An explicit replication construction and finite examples distinguish this completeness from replication at the original prices.
Comments22 pages
