Stochastic sewing lemma on Wasserstein space

TL;DR

This paper extends the stochastic sewing lemma to the Wasserstein space of probability measures, enabling the construction of limit flows and applying it to law-dependent jump SDEs for existence and uniqueness of solutions.

Contribution

It develops a Wasserstein space version of the stochastic sewing lemma, enhancing the classical result and applying it to law-dependent jump SDEs for weak existence and law uniqueness.

Findings

Constructed a limit flow of probability measures from a doubly indexed family of maps.

Provided weak existence results for law-dependent jump SDEs.

Established uniqueness of marginal laws for these SDEs.

Abstract

The stochastic sewing lemma recently introduced by Le~(2020) allows to construct a unique limit process from a doubly indexed stochastic process that satisfies some regularity. This lemma is stated in a given probability space on which these processes are defined. The present paper develops a version of this lemma for probability measures: from a doubly indexed family of maps on the set of probability measures that have a suitable probabilistic representation, we are able to construct a limit flow of maps on the probability measures. This result complements and improves the existing result coming from the classical sewing lemma. It is applied to the case of law-dependent jump SDEs for which we obtain weak existence result as well as the uniqueness of the marginal laws.