Probabilistic models and statistics for electronic financial markets in the digital age

ArXiv ID: 2406.07388 “View on arXiv”

Authors: Unknown

Abstract

The scope of this manuscript is to review some recent developments in statistics for discretely observed semimartingales which are motivated by applications for financial markets. Our journey through this area stops to take closer looks at a few selected topics discussing recent literature. We moreover highlight and explain the important role played by some classical concepts of probability and statistics. We focus on three main aspects: Testing for jumps; rough fractional stochastic volatility; and limit order microstructure noise. We review jump tests based on extreme value theory and complement the literature proposing new statistical methods. They are based on asymptotic theory of order statistics and the Rényi representation. The second stage of our journey visits a recent strand of research showing that volatility is rough. We further investigate this and establish a minimax lower bound exploring frontiers to what extent the regularity of latent volatility can be recovered in a more general framework. Finally, we discuss a stochastic boundary model with one-sided microstructure noise for high-frequency limit order prices and its probabilistic and statistical foundation.

Keywords: Semimartingales, Jump Testing, Rough Stochastic Volatility, Microstructure Noise, High-Frequency Data, General Financial Markets

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats
• Why: The paper presents advanced theoretical probability and statistics, focusing on semimartingales, minimax lower bounds, and asymptotic theory of order statistics, but is a review with no code, datasets, or backtests.

  flowchart TD
    A["Research Goal<br>Statistical Methods for<br>Discrete Semimartingales"] --> B{"Methodology"}
    
    B --> C["Testing for Jumps<br>Extreme Value Theory &<br>Rényi Representation"]
    B --> D["Rough Stochastic Volatility<br>Minimax Lower Bounds &<br>Regularity Recovery"]
    B --> E["Microstructure Noise<br>Stochastic Boundary Model<br>for Limit Orders"]
    
    C --> F["Data: High-Frequency<br>Asset Returns"]
    D --> G["Data: Volatility Estimators<br>& Latent Processes"]
    E --> H["Data: Limit Order Prices<br>& Trade Data"]
    
    F --> I["Computational<br>Asymptotic Order<br>Statistics Analysis"]
    G --> I
    H --> I
    
    I --> J["Key Findings<br>Enhanced Jump Detection<br>Volatility Regularity Bounds<br>Microstructure Noise Model"]