Decision under intensity-based costs in large-scale systems

Abstract

Rational agents make most of their decisions under uncertainty, in situations where it is impossible to anticipate the full consequences of their choices, and their impacts both on their future behavior and on their external environment. This dissertation explores how agents should optimally behave in applications drawn from the energy and finance sectors using tools laying at the broader intersection of stochastic optimization, operations research, mean field games, and machine learning. Our first application looks at a problem where electricity consumers face uncertainty about their energy costs and how they can minimize them by managing their resources. In power grids, expensive infrastructure enhancements are required to improve capacity, the maximum amount the system can handle, and transmission, the infrastructure necessary to transport electricity. These costs are reflected on end-users' bills based on their demand during coincident peak events, times when the system-wide electric load is the highest. We develop a scenario generation engine that predicts the probabilities of these events and use it to find the optimal schedule of a battery to maximize on-bill savings. The second application explores the specificity of order attribution on the Toronto Stock Exchange, where brokers can choose to trade with their identity or under a generic anonymous identifier. This leads to two different sources of price impact that degrade the traded asset price permanently and temporarily. We formulate a Stochastic Differential Game for the optimal execution problem of a finite number of brokers and solve the associated Mean Field Game (MFG) with common noise to analyze the strategic component of identity-optionality. Finally, the third application investigates the turnpike phenomenon, the propensity for the solutions of some optimal control problems defined over a long time horizon to spend most of their time near a stationary state solving the infinite time version of the optimal control problem. We review existing turnpike estimates for specific classes of MFGs and establish new results for a linear-quadratic MFG. We build upon the Deep Galerkin Method (DGM) to solve MFGs numerically through a “turnpike-accelerated” version, which incorporates such estimates into the loss function to obtain more accurate solutions more efficiently.