This paper studies an equity market of stochastic dimension, where the number of assets fluctuates over time. In such a market, we develop the fundamental theorem of asset pricing, which provides the equivalence of the following statements: (i) there exists a supermartingale numéraire portfolio; (ii) each dissected market, which is of a fixed dimension between dimensional jumps, has locally finite growth; (iii) there is no arbitrage of the first kind; (iv) there exists a local martingale deflator; (v) the market is viable. We also present the optional decomposition theorem, which characterizes a given nonnegative process as the wealth process of some investment-consumption strategy. Furthermore, similar results still hold in an open market embedded in the entire market of stochastic dimension, where investors can only invest in a fixed number of large capitalization stocks. These results are developed in an equity market model where the price process is given by a piecewise continuous semimartingale of stochastic dimension. Without the continuity assumption on the price process, we present similar results but without explicit characterization of the numéraire portfolio.

中文翻译：

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随机维度市场中的套利理论

本文研究了随机维度的股票市场，其中资产数量随时间波动。在这样的市场中，我们提出了资产定价的基本定理，该定理提供了以下陈述的等价性：（i）存在超鞅计值投资组合；(ii) 每个被剖析的市场在维度跳跃之间具有固定维度，局部增长有限；(iii) 不存在第一类套利；(iv) 存在局部鞅平减指数；(v) 市场是可行的。我们还提出了可选分解定理，它将给定的非负过程描述为某种投资-消费策略的财富过程。此外，类似的结果仍然适用于嵌入整个随机维度市场的开放市场，投资者只能投资固定数量的大盘股。这些结果是在股票市场模型中得出的，其中价格过程由随机维度的分段连续半鞅给出。如果没有价格过程的连续性假设，我们会得出类似的结果，但没有对计价投资组合进行明确的描述。