Journal of Applied Mathematics and Stochastic AnalysisVolume 2005 (2005), Issue 2, Pages 97-118doi:10.1155/JAMSA.2005.97

# Lakdere Benkherouf1 and Susan Pitts2

1Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khodh 123, Oman
2Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

Received 29 May 2004; Revised 16 September 2004

Copyright © 2005 Lakdere Benkherouf and Susan Pitts. 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.

#### Abstract

This paper is concerned with optimal strategies for drilling in an oil exploration model. An exploration area contains n1 large and n2 small oilfields, where n1 and n2 are unknown, and represented by a two-dimensional prior distribution π. A single exploration well discovers at most one oilfield, and the discovery process is governed by some probabilistic law. Drilling a single well costs c, and the values of a large and small oilfield are v1 and v2 respectively, v1>v2>c>0. At each time t=1,2,, the operator is faced with the option of stopping drilling and retiring with no reward, or continuing drilling. In the event of drilling, the operator has to choose the number k, 0km (m fixed), of wells to be drilled. Rewards are additive and discounted geometrically. Based on the entire history of the process and potentially on future prospects, the operator seeks the optimal strategy for drilling that maximizes the total expected return over the infinite horizon. We show that when ππ in monotone likelihood ratio, then the optimal expected return under prior π is greater than or equal to the optimal expected return under π. Finally, special cases where explicit calculations can be done are presented.