# Memory and Anticipation: Two main theorems for Markov regime-switching stochastic processes

## TL;DR

This paper establishes foundational theorems for Markov regime-switching stochastic processes, including existence-uniqueness of solutions and duality relations, to support future theoretical and applied research.

## Contribution

It introduces two main theorems: an existence-uniqueness result for SDDEJRs and a duality between SDDEJRs and ABSDEJRs, advancing the mathematical framework for regime-switching models.

## Key findings

- Proved existence and uniqueness of solutions for SDDEJRs.
- Established duality between SDDEJRs and ABSDEJRs.
- Provided theoretical basis for future applications in time-delayed and time-advanced models.

## Abstract

We present two main theorems for stochastic processes with a Markov regime-switching model. First, we work on an existence-uniqueness theorem for a Stochastic Differential Delay Equation with Jumps and Regimes (SDDEJRs). Then, we provide the duality between an SDDEJR and an Anticipated Backward Stochastic Differential Equation with Jumps and Regimes (ABSDEJRs). Our goal is to provide two technical and basis theorems for the future theoretical and applied developments of time-delayed and time-advanced models.