Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 82538, 19 pages

A singular control problem with an expected and a pathwise ergodic performance criterion

Andrew Jack and Mihail Zervos

Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom

Received 13 May 2005; Revised 12 April 2006; Accepted 13 April 2006

Copyright © 2006 Andrew Jack and Mihail Zervos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the problem of controlling a general one-dimensional Itô diffusion by means of a finite-variation process. The objective is to minimise a long-term average expected criterion as well as a long-term pathwise criterion that penalise deviations of the underlying state process from a given nominal point as well as the expenditure of control effort. We solve the resulting singular stochastic control problems under general assumptions by identifying an optimal strategy that is explicitly characterised.