**Title**: Games in Energy Markets

**Time and location**: Friday, 10/20, 2:00pm at South Hall 5421 (StatLab)

**Committee**:

Dr. Mike Ludkovski (chair)

Dr. Tomoyuki Ichiba

Dr. Jean-Pierre Fouque

**Abstract**:

We study energy markets in the framework of stochastic differential game. Two types of energy producers are considered: exhaustible producers and renewable producers. An exhaustible producer produces energy with exhaustible resources, such as oil. The resource reserves of each exhaustible producer diminish due to production, and also get replenished with costly effort to explore for new resources. This exploration activity is modeled through a controlled point process that leads to stochastic increments to reserves level. A renewable producer uses renewable resources, such as solar power, to produce energy. The renewable resources are infinite, but costly in production. Each producer chooses optimal controls of production quantity and exploration effort (exhaustible producers only), in order to maximize individual profit that equals his quantity of production multiplied by market price, minus costs of production and exploration. The producers interact with each other through the energy price that is a function of aggregate production. We aim to study the dynamic games between energy producers in terms of Nash equilibria.

In the first part, we study the game between an exhaustible producer and a renewable producer under stochastic demand that switches between different regimes. Our aim is to study this game of the exhaustible producer with a renewable producer in terms of dynamic Nash equilibria. We do computational analysis on the coupled systems of Hamilton-Bellman-Isaacs (HJB-I) differential equations derived by dynamic programming approach. We study how the regime changes and the relative cost of production, which is a proxy for market competitiveness, affect game equilibria, and compare with the case of deterministic demand. A novel feature driven by stochasticity of demand is that production may shut down during low demand to conserve reserves.

In the second part, we study energy markets with a continuum of homogeneous exhaustible producers. We employ a mean field game 1 approach to solve for a Markov Nash equilibrium. We develop numerical schemes to solve the resulting system of partial differential equations: a backward Hamilton-Jacobi-Bellman (HJB) equation for the value function of a representative producer and a forward transport equation for the distribution of the reserves levels among all producers. A time-stationary formulation is also explored, as well as the fluid limit where exploration becomes deterministic. Through the mean field formulation, we aim to study equilibrium production quantities and reserves distribution in energy markets with a large population of competing producers. Particularly, we want to understand how exploration activities affect the long-term market organization, and how the exploration uncertainty permeates the solution.