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Arbitrage theory in a market of stochastic dimension

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  • Erhan Bayraktar
  • Donghan Kim
  • Abhishek Tilva

Abstract

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.

Suggested Citation

  • Erhan Bayraktar & Donghan Kim & Abhishek Tilva, 2024. "Arbitrage theory in a market of stochastic dimension," Mathematical Finance, Wiley Blackwell, vol. 34(3), pages 847-895, July.
    • Handle: RePEc:bla:mathfi:v:34:y:2024:i:3:p:847-895
    DOI: 10.1111/mafi.12418
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    Cited by:

    1. Erhan Bayraktar & Donghan Kim & Abhishek Tilva, 2024. "Quantifying dimensional change in stochastic portfolio theory," Mathematical Finance, Wiley Blackwell, vol. 34(3), pages 977-1021, July.

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