On some auto-induced regime switching double threshold glued diffusions

Regime switching processes are usually defined with an external random sourcedriving the regime changes. In this paper we define and study a regime switchingdiffusion considering two thresholds, and regime switching occurring, by a change inthe diffusion drift and volatility, whenever the trajectory touches the upper thresholdafter having crossed, or touched, the lower threshold or touches the lower thresholdafter having crossed, or touched, the upper threshold. We develop an estimationprocedure for the thresholds and for the regime parameters of the diffusions. Weshow that a generalized Black-Scholes model with the regime switching diffusionas the law of the risky asset is arbitrage free and complete under an additionalhypothesis on the diffusion coefficients of the two regimes diffusions.