A C\`adl\`ag Rough Path Foundation for Robust Finance

Abstract

Using rough path theory, we provide a pathwise foundation for stochastic It\^o 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 c\`adl\`ag paths, which is shown to imply the existence of a c\`adl\`ag rough path and of quadratic variation in the sense of F\"ollmer. 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.