Convergence for Reflecting Boundaries, Singular Control and Ergodic Cost Problems

Abstract

The development of the convergence proofs of Chapter 10 is continued, but applied to the problem classes of Chapters 7 and 8. The reflecting boundary and discounted cost problem is covered in Section 11.1. The primary mathematical difficulty with which we must contend is the proof of tightness of the “reflecting process.” The problem is avoided by use of a time rescaling method, under which all the processes are tight. After proving the weak convergence of the rescaled processes and characterizing the limits, the rescaling is inverted to obtain the desired results. This “inversion” is possible due to the conditions imposed on the allowable reflection directions. The time rescaling idea appears to be a rather powerful tool.