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On a multidimensional oil exploration problem


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Authors

Benkherouf, Lakdere 
Pitts, Susan 

Abstract

jats:pThis paper is concerned with optimal strategies for drilling in an oil exploration model. An exploration area contains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n1">mml:mrowmml:msubmml:min</mml:mi>mml:mn1</mml:mn></mml:msub></mml:mrow></mml:math> large and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n2">mml:mrowmml:msubmml:min</mml:mi>mml:mn2</mml:mn></mml:msub></mml:mrow></mml:math> small oilfields, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n1">mml:mrowmml:msubmml:min</mml:mi>mml:mn1</mml:mn></mml:msub></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n2">mml:mrowmml:msubmml:min</mml:mi>mml:mn2</mml:mn></mml:msub></mml:mrow></mml:math> are unknown, and represented by a two-dimensional prior distribution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="π"><mml:mi mathvariant="bold">π</mml:mi></mml:math>. A single exploration well discovers at most one oilfield, and the discovery process is governed by some probabilistic law. Drilling a single well costs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c">mml:mic</mml:mi></mml:math>, and the values of a large and small oilfield are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v1">mml:msubmml:miv</mml:mi>mml:mn1</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v2">mml:msubmml:miv</mml:mi>mml:mn2</mml:mn></mml:msub></mml:math> respectively, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Misplaced &v_1&gt;v_2&gt;c&gt;0v_1&gt;v_2&gt;c&gt;0">mml:msubmml:miv</mml:mi>mml:mn1</mml:mn></mml:msub>mml:mo></mml:mo>mml:msubmml:miv</mml:mi>mml:mn2</mml:mn></mml:msub>mml:mo></mml:mo>mml:mic</mml:mi>mml:mo></mml:mo>mml:mn0</mml:mn></mml:math>. At each time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t=1,2,…">mml:mit</mml:mi>mml:mo=</mml:mo>mml:mn1</mml:mn>mml:mo,</mml:mo>mml:mn2</mml:mn>mml:mo,</mml:mo>mml:mo…</mml:mo></mml:math>, 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k">mml:mik</mml:mi></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0≤km">mml:mn0</mml:mn>mml:mo≤</mml:mo>mml:mik</mml:mi>mml:mo≤</mml:mo>mml:mim</mml:mi></mml:math> (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m">mml:mim</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ππ"><mml:mi mathvariant="bold">π</mml:mi>mml:mo≻</mml:mo>mml:msup<mml:mi mathvariant="bold">π</mml:mi>mml:mo′</mml:mo></mml:msup></mml:math> in monotone likelihood ratio, then the optimal expected return under prior <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="π"><mml:mi mathvariant="bold">π</mml:mi></mml:math> is greater than or equal to the optimal expected return under <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="π">mml:msup<mml:mi mathvariant="bold">π</mml:mi>mml:mo′</mml:mo></mml:msup></mml:math>. Finally, special cases where explicit calculations can be done are presented.</jats:p>

Description

Keywords

4901 Applied Mathematics, 49 Mathematical Sciences

Journal Title

Journal of Applied Mathematics and Stochastic Analysis

Conference Name

Journal ISSN

1048-9533
1687-2177

Volume Title

Publisher

Hindawi Limited