# Comparison Theorem for Distribution Dependent Neutral SFDEs

## TL;DR

This paper establishes existence, uniqueness, and order preservation conditions for distribution dependent neutral stochastic functional differential equations, extending known results and providing new necessary conditions.

## Contribution

It introduces new necessary and sufficient conditions for order preservation in distribution dependent neutral SFDEs, extending previous distribution independent results.

## Key findings

- Proved existence and uniqueness of strong solutions.
- Established conditions for order preservation.
- Extended results to distribution dependent case.

## Abstract

In this paper, the existence and uniqueness of strong solutions to distribution dependent neutral SFDEs are proved. We give the conditions such that the order preservation of these equations holds. Moreover, we show these conditions are also necessary when the coefficients are continuous. Under sufficient conditions, the result extends the one in the distribution independent case, and the necessity of these conditions is new even in distribution independent case.