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Register a new userA Sobolev rough path extension theorem via regularity structures
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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 theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.
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We show that every $mathbb{R}^d$-valued Sobolev path with regularity $alpha$ and integrability $p$ can be lifted to a Sobolev rough path in the sense of T. Lyons provided $alpha >1/p>0$. Moreover, we prove the existence of unique rough path lifts which are optimal w.r.t. strictly convex functionals among all possible rough path lifts given a Sobolev path. As examples, we consider the rough path lift with minimal Sobolev norm and characterize the Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. to a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.
We introduce the space of rough paths with Sobolev regularity and the corresponding concept of controlled Sobolev paths. Based on these notions, we study rough path integration and rough differential equations. As main result, we prove that the solution map associated to differential equations driven by rough paths is a locally Lipschitz continuous map on the Sobolev rough path space for any arbitrary low regularity $alpha$ and integrability $p$ provided $alpha >1/p$.
This work is devoted to a vast extension of Sanovs theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of i.i.d. samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis-Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.
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. Without the assumption of any underlying probabilistic model, we prove pathwise Master formulae analogous to those of classical SPT, describing the growth of wealth processes associated to functionally generated portfolios relative to the market portfolio. We show that the appropriately scaled asymptotic growth rate of a far reaching generalization of Covers universal portfolio based on controlled paths coincides with that of the best retrospectively chosen portfolio within this class. We provide several novel results concerning rough integration, and highlight the advantages of the rough path approach by considering (non-functionally generated) log-optimal portfolios in an ergodic It^o diffusion setting.
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.
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