Optimal switching strategy of a mean-reverting asset over multiple regimes

We solve optimal iterative three-regime switching problems with transaction costs, with investment in a mean-reverting asset that follows an Ornstein–Uhlenbeck process and find the explicit solutions. The investor can take either a long, short or square position and can switch positions during the period. Modeling the short sales position is necessary to study optimal trading strategies such as the pair trading. Few studies provide explicit solutions to problems with multiple (more than two) regimes (states). The value function is proved to be a unique viscosity solution of a Hamilton–Jacobi–Bellman variational inequality (HJB-VI). Multiple-regime switching problems are more difficult to solve than conventional two-regime switching problems, because they need to identify not only when to switch, but also where to switch. Therefore, multiple-regime switching problems need to identify the structure of the continuation/switching regions in the free boundary problem for each regime. If the number of the states NN is two, only two regions have to be identified, but if N=3N=3, NP2=6NP2=6 regions have to be detected. We identify the structure of the switching regions for each regime using the theories related to the viscosity solution approach.