Six simple piecewise deterministic oil production models from economics are discussed by using solution tools that are available in the theory of piecewise deterministic optimal control problems.

Six simple piecewise deterministic models for optimal oil-owner behavior are presented. Their central property is sudden jumps in states. The aim of this paper is to show in admittedly exceedingly simple models how available tools for piecewise deterministic models, namely, the HJB equation and the maximum principle, can be used to solve these models analytically. We are looking for solutions given by explicit formulas. That can only be obtained if the models are simple enough. The models may be too simple to be of much interest in themselves, but they can provide some intuition about features optimal solutions may have in more complicated models.

Piecewise deterministic models have been used a number of times in economic problems in the literature; some few scattered references are given that contain such applications [

In all models below, an unbounded number of jumps in the state can occur at times

Consider the optimization problem

The interpretation of the model is that

Let us solve problem (

The adjoint equation (see

Now, for

Write

Write

We are now going to replace the bequest function

If we drop the term

Let us discuss a little more the value of

Then,

Now, with probability 1 an infinite number of jumps occur in

We may assume that the jumps are stochastic, that is, that

Note that we must assume that we have a deal with the bank in which our wealth is placed that it accepts the above behavior. That is, before time 0, we have got an acceptance for the possibility of operating with this type of admissible solutions, which means that only in expectation we leave a wealth in the bank

Consider now the case where

(This model Is related to exercise 4.1 in Øksendal and Sulem [

Consider the following problem:

In contrast to problem (

It is easy to see that the current value function

Now, assume first, for some

Now,

Let

We can show that the

If all

In this model, the physical volume of oil production is constant

Let us use the extremal method (see the appendix) for solving this problem. In what follows, it is guessed that adjoint functions do not depend on arbitrarily given initial points (which in the appendix are denoted by

Next, maximum of the Hamiltonian

Now,

Note that, for any

If

In this example,

Let

To show the existence of a solution of (

For any admissible

Define

Two very simple models discussed below contain the feature that the owner can influence the chance of discovery but that it is costly to do so. In the first one, the intensity of discoveries

Consider the problem

Let us try the proposal that the current value optimal value function

Often, it is the case that the best fields are exploited first hence,

Still, trivially condition

Consider the problem

All fields found are of the same size

Let us try the proposal that the current value function

In Models 1, 2, 5, and 6, oil finds are made at stochastic points in time; in Models 3 and 4, it is the price of oil that changes at stochastic points in time. In Model 1, we operate with the constraint

In Models 3 and 4, the oil price exhibits sudden stochastic jumps. In Model 3, the rate of oil production is constant, but income earned (as well as interest) is placed in a bank after subtraction of consumption. In Model 4, income earned, after subtraction of consumption, is reinvested in the oil firm to increase production. In Model 4, the optimal control is stochastic; it depends on whether the current price is high or low. In Model 3, the control is deterministic, and it depends only on the expectation

In the extremely simple Models 5 and 6, the frequency of oil finds is not fixed but influenced by a control. In Model 5, the current frequency (or intensity) is determined by how much money is put into search at that moment in time. In the simplest case considered in Model 5, a find today does not influence the possibility of making equally sized discoveries tomorrow. Then it, is not unreasonable that the optimal control (which equals

Consider the problem

Assume that the five given functions

Assume that there exist functions

See pages 147 and 155 (and for a proof, see page 168) in [

In case of restrictions of the form

If there are terminal conditions in the problem, replace

If

For

(This is the infinite horizon current value form of (

Frequently, to find

The entities

So far, we have rendered conditions pertaining to a free terminal state problem. If there are hard terminal conditions (hard = holding a.s.) or soft terminal conditions (holding in expectation), then certain transversality conditions have to be satisfied by the

Assume now that entities

If all the entities

In the end constrained case,

When

The author is very grateful for useful comments received from a referee that made it possible to improve the exposition and remove some typographical errors.