Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article


Doctor of Philosophy


Statistics and Actuarial Sciences


Davison, Matt

2nd Supervisor

Escobar-Anel, Marcos



In this thesis, we study two continuous-time optimal control problems. The first describes competition in the energy market and the second aims at robust portfolio decisions for commodity markets. Both problems are approached via solutions of Hamilton-Jacobi-Bellman (HJB) and HJB-Isaacs (HJBI) equations.

In the energy market problem, our target is to maximize profits from trading crude oil by determining optimal crude oil production. We determine the optimal crude oil production rate by constructing a differential game between two types of players: a single finite-reserve producer and multiple infinite-reserve producers. We extend the deterministic unbounded-production model and stochastic monopolistic game to bounded-production and construct an $N$-player stochastic game using analytical and numerical solutions to the corresponding HJB equation. In this way, we compute the optimal strategies of oil production for four stylized players. As an example, applying the game-theory model above, we construct a deterministic and a stochastic differential-game model between four countries, and compare the real production and the forecast production in order to test the accuracy of the model.

In the robust portfolio optimization problem, we assume the investor allocates funds among a bond, a bank account, and a commodity that either pays a mean-reverting convenience yield, or follows an exponential Ornstein-Uhlenbeck (OU) process. In our settings, the interest rate of the bond follows a Vasicek model. We optimize the expected utility of terminal wealth, solving the corresponding HJBI equation via an exponential affine ansatz, which can be used to generate an optimal portfolio strategy. As part of our study, we fit our model to prices of crude oil, gold, copper and interest rates, leading to a meaningful empirical analysis. We concluded from the suboptimal analysis that the mis-specification of parameters and incompleteness of market lead to severe wealth-equivalent losses.

Summary for Lay Audience

This thesis addresses two problems in the theory of commodity markets, unified by the Hamilton Jacobi Bellman (HJB) stochastic optimal control methodology employed and solve associated HJB partial differential equation systems.

The first problem involves applying differential game approaches to model the production strategy of energy producers. In the game, each player determines their own production rate schedule, which affects the world energy market supply-demand relationship and hence price. We do this both in a deterministic system and in a system in which energy demand has a stochastic driver. In both cases we consider the impact of participant production and profit bounds.

The second topic treated is the robust optimization of portfolios which include commodity assets. Here an investor solves for an optimal wealth allocation in order to maximize their expected terminal wealth, in a worst-case expected return scenario. The worst-case scenario is due to ambiguity in asset return parameters. We also compute the scale of losses by, alternatively, ignoring the ambiguity but considering a case in which parameters are incorrectly estimated.

These studies will be helpful for oil or other resource producing blocs to select their production strategies and for ambiguity-averse investors to determine their optimal investment strategy.

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

Available for download on Wednesday, June 14, 2023