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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 Follmer. 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 Covers universal portfolio are admissible integrands, and that Property (RIE) is satisfied by both (Young) semimartingales and typical price paths.
We investigate rough differential equations with a time-dependent reflecting lower barrier, where both the driving (rough) path and the barrier itself may have jumps. Assuming the driving signals allow for Young integration, we provide existence, uni
Based on a rough path foundation, we develop a model-free approach to stochastic portfolio theory (SPT). Our approach allows to handle significantly more general portfolios compared to previous model-free approaches based on Follmer integration. With
Using Dupires notion of vertical derivative, we provide a functional (path-dependent) extension of the It^os formula of Gozzi and Russo (2006) that applies to C^{0,1}-functions of continuous weak Dirichlet processes. It is motivated and illustrated b
We show that every $mathbb{R}^d$-valued Sobolev path with regularity $alpha$ and integrability $p$ can be lifted to a Sobolev rough path provided $alpha < 1/p<1/3$. The novelty of our approach is its use of ideas underlying Hairers reconstruction the
Empirical risk minimization over classes functions that are bounded for some version of the variation norm has a long history, starting with Total Variation Denoising (Rudin et al., 1992), and has been considered by several recent articles, in partic