Optimal Oil-Owner Behavior in Piecewise Deterministic Models

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. 1. Introduction 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 [1–4]. I have not been able to find references directly concerned with piecewise deterministic oil production problems. For different probability structures, and for discrete time, a host of related problems has been discussed in the literature; references to such literature have been left out, with one exception. Problems of control of jump diffusions, see [5], encompass piecewise deterministic problems, and some problems appearing in [5] are related to the ones discussed below. A classic reference to piecewise deterministic control problems is [2]. In all models below, an unbounded number of jumps in the state can occur at times , and, when is given, is exponentially distributed in (all independent). Sometimes, the size of the jumps is influenced by stochastic variables . Let . At time , we imagine that the control values chosen can be allowed to depend on what has happened, that is, on , for , but not on future events, that is, , for which . Such controls (written ) are called nonanticipative. Corresponding state solutions denoted by are then also nonanticipative. (A general set-up, with further explanations, is given in Appendix.) Frequently below, will be the wealth of the oil owner. In infinite horizon economic models, the weakest terminal condition that is natural to apply is an expectational no-Ponzi-game condition; namely, , where is the discount factor. Note that some stronger conditions will be used in some of the models presented in the sequel. 2. Model 1 Consider the optimization problem where is the control, subject