On some auto-induced regime switching double threshold glued diffusions Manuel L. Esqu´ ıvel *† Pedro P. Mota ‡§ December 22, 2011 As an heartily and humble token of immense admiration and everlasting gratitude, this work is dedicated by the first author to Jean-Pierre Kahane, his friend, PhD adviser and patient mathematical mentor for more than 20 years, from whom he first learned, at Orsay lectures, the surprising and beautiful level sets behavior of fractional (and usual) Brownian processes. Abstract Regime switching processes are usually defined with an external random source driving the regime changes. In this paper we define and study a regime switching diffusion considering two thresholds, and regime switching occurring, by a change in the diffusion drift and volatility, whenever the trajectory touches the upper threshold after having crossed, or touched, the lower threshold or touches the lower threshold after having crossed, or touched, the upper threshold. We develop an estimation procedure for the thresholds and for the regime parameters of the diffusions. We show that a generalized Black-Scholes model with the regime switching diffusion as the law of the risky asset is arbitrage free and complete under an additional hypothesis on the diffusion coefficients of the two regimes diffusions. 1 Introduction Let us describe some motivation for the study presented in this paper, with a tentative modeling of a simple phenomena. Consider a closed room containing a heating device equipped with a thermostat; whenever the temperature drops below some given threshold c the heating device starts delivering heat until the temperature gets over the threshold. Due to some thermal inertia of the heating device, the temperature will continue to * Departamento de Matem´ atica, Faculdade de Ciˆ encias e Tecnologia da Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal. † This work was partially supported by CMA/FCT/UNL, under the project PEst- OE/MAT/UI0297/2011. ‡ Departamento de Matem´ atica, Faculdade de Ciˆ encias e Tecnologia da Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal. § This work was partially supported by CMA/FCT/UNL, under the project PEst- OE/MAT/UI0297/2011. 1