Memory and Anticipation: Two Main Theorems for Markov Regime-Switching Stochastic Processes

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 fundamental theorems for the future theoretical and applied developments of time-delayed and time-advanced models.