Rough Paths for Pathwise Integration and Robust Trading Strategies

Summary

This theoretical work develops a pathwise basis for stochastic Itô integration using rough path theory, with the aim of covering common trading strategies and financial models under uncertainty. It introduces Property (RIE) for càdlàg paths and shows that this condition yields a càdlàg rough path and quadratic variation in the Föllmer sense. The associated rough integrals are constructed as limits of left-point Riemann sums along suitable partitions, including for integrands that are not of gradient type.

The framework provides access to rough path stability estimates and establishes that path-dependent functionally generated strategies and Cover's universal portfolio are admissible integrands. The authors also show that the property holds for Young semimartingales and typical price paths. This is a mathematical foundation rather than an empirical trading test: it does not report strategy returns or demonstrate out-of-sample profitability. Practical application depends on the stated path conditions and on the assumptions behind the integration results.

Key ideas

• Property (RIE) is introduced as a condition on càdlàg paths that supports a rough path construction.
• Under this condition, quadratic variation exists in the Föllmer sense.
• Rough integrals are obtained as limits of left-point Riemann sums along suitable partitions.
• The framework accommodates non-gradient integrands and offers rough path stability estimates.
• Functionally generated strategies and Cover's universal portfolio are shown to be admissible integrands.

Tags

# A Càdlàg Rough Path Foundation for Robust Finance


# A Càdlàg Rough Path Foundation for Robust Finance









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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.