Escape Dynamics: A Continuous Time Approximation

In this paper we provide an explicit characterization of the escape dynamics for the Phellps problem of government controlling inflation with adaptive learning of the approximate Phillips curve, alternative to the one considered by Cho, Williams and Sargent (2002) (CWS in sequel). Our approach is based on approximating the discrete-time stochastic recursive algorithm, which describes dynamics with learning in this problem, by the limiting continuous-time process (diffusion). We subsequently characterize the escape dynamics (escape time and dominant escape path) for this limit process. CWS derive the characteristics of the escape dynamics for the original discrete time stochastic recursive algorithm using extension of the Freidlin and Wentzell (1998) large deviations theory by Dupuis and Kushner (1989). This theory allows one to derive the escape time and the dominant escape path for discrete time models with bounded shocks, but not unbounded (Gaussian) shocks. In the latter case only the upper bound of probability of large deviations can be derived, while both upper and lower bounds on this probability are necessary to derive the escape times and dominant escape path. Switching to continuous time approximation allows us to avoid the problem of unboundedness of shocks in discrete time. It allows us to use well-developed theory of large deviations for continuous time processes to characterize fully the escape dynamics with unbounded continuous-time shocks by using Euler and Hamilton-Jacobi differential equations.