OPTIMAL SWITCHING UNDER A REGIME-SWITCHING MODEL WITH TWO-TIME-SCALE MARKOV CHAINS

This paper is concerned with a probabilistic approach to an optimal switching problem. The dynamics of the system consists of a set of diffusions coupled by a finite-state Markov chain. It is shown that the corresponding value function can be given in terms of the solution of an oblique reflected backward stochastic differential equation with a Markov chain. In many applications, the underlying Markov chain exhibits two-time-scale structure. In this case, the value function for the original problem is shown to converge to the value function of a limit problem as the fluctuation rate shrinks to zero. The main advantage of this two-time-scale approach is the reduction of dimensionality. The limit problem is much easier to solve, and its optimal switching solution leads to approximate solutions to the original problem. Finally, a numerical example is provided to demonstrate the convergence result.