Bridging rough semimartingales and rough stochastic differential equations

The theory of rough stochastic differential equations (RSDEs) initiated by Friz, Hoquet and Lê in 2021 provides an intrinsic solution theory for stochastic dynamics simultaneously driven by a martingale noise N and a rough path X. Their approach crucially relies on a suitable modification of Lê’s stochastic sewing lemma, which comes with certain moment assumptions on the noise N. While this is usually fine in a continuous setting, the high moment assumptions cause considerable problems for establishing a useful rough stochastic calculus when considering e.g. N to be a general Lévy-type martingale with jumps. For example, an Itô formula for RSDEs driven by such noise is not available in the FHL framework.

In parallel, Friz and Zorin-Kranich in 2023 introduced the notion of rough semimartingales (RSMs), sums of martingales and controlled rough paths, which can be shown to be isomorphic to suitable spaces of Gubinelli’s controlled rough paths. Using this connection one can then build a rough stochastic calculus based on RSMs, which is (almost) moment-free and thus does not face the above issue.

In this talk I will show how we construct the RSM-based rough stochastic calculus and how this can be applied to solutions of càdlàg RSDEs.  We show that under suitable coefficient assumptions standard for RSDE well-posedness — and with no moment condition on N beyond those that well-posedness already requires — every RSDE solution is also an RSM. The calculus therefore applies to such solutions, and in particular yields the Itô formula that is unavailable in the FHL framework.  

 

Based on joint works with Peter K. Friz, Arnulf Jentzen, Jost Pieper and Jian Song.