论文标题
随机维度市场中的套利理论
Arbitrage theory in a market of stochastic dimension
论文作者
论文摘要
本文研究了随机维度的股票市场，其中资产数量随着时间而波动。在这样的市场中，我们开发了资产定价的基本定理，该定理提供了以下陈述的等效性：（i）有一个超级马丁·纳米尔（SupermartingaleNuméraire）投资组合； （ii）每个在尺寸跳跃之间具有固定维度的解剖市场具有局部有限的增长； （iii）没有第一种套利； （iv）有一个当地的马丁加尔缩写器； （v）市场是可行的。我们还提出了可选的分解定理，该定理将给定的非负过程描述为某些投资消费策略的财富过程。此外，类似的结果仍然存在于整个随机维度市场中的公开市场中，投资者只能投资固定数量的大型资本化股票。这些结果是在股票市场模型中开发的，在股票市场模型中，价格过程是由随机维度的分段连续的半明星给出的。如果没有关于价格过程的连续性假设，我们提出了相似的结果，但没有明确表征Numéraire产品组合。
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.
