ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF SYSTEMS OF KOLMOGOROV BACKWARD EQUATIONS FOR TWO-TIME-SCALE SWITCHING DIFFUSIONS

This work is concerned with systems of coupled partial differential equations (known as Kolmogorov backward equations) for continuous-time Markov processes featuring in the coexistence of continuous dynamics and discrete events. Arising from state-dependent switching diffusions, distinct from the usual Markovian regime-switching systems, the generator of the switching component depends on the continuous state. One of the main ingredients of our models is the two-time-scale formulation. In contrast to the work on Kolmogorov forward equations in the existing literature, new techniques are developed in this paper. Although they originate from probabilistic models, the methods are analytic. Two classes of models, namely, fast-switching systems and fast-diffusion systems, are treated. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations. These asymptotic series are rigorously justified and error bounds are obtained.