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A Càdlàg Rough Path Foundation for Robust Finance

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

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.

A Càdlàg Rough Path Foundation for Robust Finance

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.

Paper Structure

This paper contains 17 sections, 18 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Rough integration under Property (RIE)
  3. Basic notation
  4. Càdlàg rough path theory and Property (RIE)
  5. The rough integral as a limit of Riemann sums
  6. Link to Föllmer integration
  7. Functionally generated trading strategies and their generalizations
  8. Price paths and admissible strategies
  9. Functionally generated trading strategies
  10. Path-dependent functionally generated trading strategies
  11. Cover's universal portfolio
  12. Functionally generated portfolios from stochastic portfolio theory
  13. Semimartingales and typical price paths satisfy Property (RIE)
  14. Semimartingales
  15. Generalized semimartingales
  16. ...and 2 more sections

Key Result

Proposition 2.4

Let $\mathbf{X} = (X,Z,\mathbb{X}) \in \mathscr{V}^p$ be a càdlàg rough path, and let $(F,F') \in \mathcal{V}^{q,r}_Z$ and $(G,G') \in \mathcal{V}^{q,r}_X$ be controlled paths with respect to $Z$ and $X$, respectively, with remainders $R^F$ and $R^G$. Then, for each $t\in [0,T]$, the limit exists along every sequence of partitions $\mathcal{P}$ of the interval $[0,t]$ with mesh size $|\mathcal{P}

Theorems & Definitions (55)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • Remark 2.9
  • ...and 45 more