Journal of Applied Mathematics and Stochastic Analysis
Volume 2005 (2005), Issue 2, Pages 97-118
doi:10.1155/JAMSA.2005.97
On a multidimensional oil exploration problem
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,
0≤k≤m
(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
π≻π′ 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.