Auto and externally induced regime switching diffusions

In the current literature we can find mainly two approaches to the SDE regime switching modeling. The traditional one, the externally induced regime switching diffusions is described by the switching being derived from a separate continuous time Markov process, with a finite, or denumerable, state space-indexing the regimes-the random times of the regime switches being exactly the jump times of the finite valued Markov process. There is a first alternative approach in which the regime switching occurs whenever the trajectory enters in some prescribed region on the state space; the regions we consider will be mainly open intervals defined by unknown thresholds for the trajectories; thresholds that, in principle, should also be estimated. In this approach the partitioning of the the state space is already defined in the drift, and volatility of the SDE. In a second alternative approach the switching occurs in a random way but at some random times defined when the trajectories hit some prescribed thresholds, that again, must be estimated. We may designate these two alternative approaches as auto-induced regime switching diffusions as there is no external noise source to force the switching occurrence. We prove a generalization of an existence result of the existence of auto-induced regime switching SDE solutions for irregular coefficients and a result that encompasses some of the cases of both externally and auto-induced regime switching SDE solutions.