Order Preservation for Path-Distribution Dependent SDEs

TL;DR

This paper establishes conditions under which path-distribution dependent stochastic differential equations preserve order, extending classical results to models where dynamics depend on both paths and distributions.

Contribution

It provides necessary and sufficient conditions for order preservation in path-distribution dependent SDEs, a significant extension beyond distribution-independent cases.

Findings

Derived necessary and sufficient conditions for order preservation.

Introduced a method to construct probability spaces based on initial distributions.

Extended classical order preservation results to more complex SDE models.

Abstract

Sufficient and necessary conditions are presented for the order preservation of path-distribution dependent SDEs. Differently from the corresponding study of distribution independent SDEs, to investigate the necessity of order preservation for the present model we need to construct a family of probability spaces in terms of the ordered pair of initial distributions.