Paper: arXiv 2212.04623

Authors: 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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly rigorous mathematical treatment extending the Fundamental Theorem of Asset Pricing to markets of stochastic dimension using advanced semimartingale theory. However, it is purely theoretical with no empirical backtesting or real-world dataset validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop FTAP in a Market of Stochastic Dimension] --> B[Data/Inputs: Piecewise Continuous Semimartingale Price Processes]
    B --> C[Methodology: Analyze Dissected Markets Between Dimensional Jumps]
    C --> D[Computational Process: Derive Optional Decomposition Theorem & Local Martingale Deflators]
    D --> E[Key Finding 1: Equivalence of No-Arbitrage NAK1, Viability, & Supermartingale Numéraire Portfolio]
    D --> F[Key Finding 2: Framework Extends to Embedded Open Markets & Jump Price Processes]