A càdlàg rough path foundation for robust finance

Allan, Andrew L.; Liu, Chong; Prömel, David J.

Abstract

Using rough path theory, we provide a pathwise foundation for stochastic Itô integration which covers most commonly applied trading strategies and mathematical models of financial markets, including those under Knightian uncertainty. To this end, we introduce the so-called property (RIE) for càdlàg paths, which is shown to imply the existence of a càdlàg rough path and of quadratic variation in the sense of Föllmer. We prove that the corresponding rough integrals exist as limits of left-point Riemann sums along a suitable sequence of partitions. This allows one to treat integrands of non-gradient type and gives access to the powerful stability estimates of rough path theory. Additionally, we verify that (path-dependent) functionally generated trading strategies and Cover’s universal portfolio are admissible integrands, and that property (RIE) is satisfied by both (Young) semimartingales and typical price paths.

Citation

Allan, A. L., Liu, C., & Prömel, D. J. (2024). A càdlàg rough path foundation for robust finance. Finance and Stochastics, 28(1), 215-257. https://doi.org/10.1007/s00780-023-00522-0

• Journal Article Type: Article
• Acceptance Date: Apr 19, 2023
• Online Publication Date: Nov 17, 2023
• Publication Date: Jan 1, 2024
• Deposit Date: May 2, 2023
• Publicly Available Date: Jan 4, 2024
• Journal: Finance and Stochastics
• Print ISSN: 0949-2984
• Electronic ISSN: 1432-1122
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 28
• Issue: 1
• Pages: 215-257
• DOI: https://doi.org/10.1007/s00780-023-00522-0
• Keywords: Semimartingale, Föllmer integration, Rough path, G11, Universal portfolio, G10, Functionally generated portfolios, 60G44, 60L20, 91G80, C50, Model uncertainty, Pathwise integration
• Public URL: https://durham-repository.worktribe.com/output/1176213