On the use of stochastic differential games against nature to ergodic control problems with unknown parameters

Abstract

In this paper, we propose an application of the so-called games against nature for giving solution to an ergodic control problem governed by a general class of Markov diffusion processes whose coefficients depend on an unknown and non-observable parameter. To this end, we assume that the values of the parameter are taken by means of ‘actions’ made by some opposite player of the controller (the nature). Then, the problem reduces to finding optimality for the controller given that the nature has chosen its best strategy. Such a control is also known as the worst case optimal control. Our analysis is based on the use of the dynamic programming technique by showing, among other facts, the existence of classical (twice differentiable) solutions of the so called Hamilton Jacobi Bellman equation. We also provide an example on economic welfare to illustrate our results.

Keywords:

• games against nature
• ergodic pay-off criterion
• controlled diffusions with unknown parameters

2010 Mathematics Subject Classification:

• 93E20
• 60J60
• 91A25

Acknowledgements

The authors wish to thank Dr. Francoise Lamnabhi-Lagarrigue, A.E. of the International Journal of Control, as well as the anonymous referees who devoted their time and effort to reviewing our work for their valuable comments.