Date of Award

May 2019

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Richard H Stockbridge

Committee Members

Chao Zhu, Gabriella Pinter, Istvan Lauko, David Spade

Keywords

non-local, non-local processes, partial integro-differential, PIDE, stochastic control, viscosity solution

Abstract

Modern electricity pricing models include a strong reversion to a long run mean and a

number of non-local operators to encapsulate the discontinuous price behavior observed in

such markets. However, incorporating non-local processes into a stochastic control problem

presents significant analytical challenges. The motivation for this work is to solve the problem

of optimal control of the burn rate for a coal-powered electricity plant. We first construct a

pricing model that is a good general representative of the class of models currently used for

electricity pricing as well as a model for the supply of fuel to the plant. Under this model,

we state the control problem of maximizing the expected discounted revenue until the first

time at which the plant runs out of fuel. Deriving the HJB equation for this control problem

results in a partial integro-differential equation, which does not t the classical theory of

viscosity solutions. Building o of work by Barles and Imbert on viscosity solutions for non-local

processes, we extend their theory to apply to non-local processes which also include a

mean-reversion component. We first show that the value function for the control problem

is a solution to this HJB equation. In our main result, we prove a comparison principle for

viscosity solutions which uses a slightly more regular structure of the non-local operators to

relax some of the assumptions of Barles and Imbert. Using this comparison principle, we

are able to show that the value function is in fact the unique solution to the HJB equation.

Thus, we have the desired result that solving the HJB equation is equivalent to solving the

control problem, giving us a direct method for finding the optimal control policy for the

electricity producer.

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