Decision under intensity-based costs in large-scale systems

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. These questions arise naturally across a wide array of industries and lead to interesting questions falling under the general scope of decision and control theory. To make decisions in the face of uncertainty, agents need to formulate and solve optimization problems. Their positing requires a mathematical description of the stochastic environment and other unknown variables, as well as the interactions between the agents with their environment. The subsequent solution consists in looking for the optimal strategy to maximize a utility function. 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, numerical analysis 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 optimally managing the resources at their disposal. In North America, power grid operators require costly infrastructure updates and construction work in order to ensure continuous electricity availability across the network. These enhancements are meant to improve capacity, the maximum amount the system can handle, and transmission, the core infrastructure necessary to transport electricity to the consumers. These costs are then reflected on end-users’ bills based on their demand during coincident peak (CP) events, time intervals in a pre-determined time frame when the system-wide electric load is the highest. We develop a scenario generation engine that predicts the probabilities of these CP events. We then use this algorithm and its variants to find the optimal schedule of a battery to reduce different costs, including coincident-peak-based ones.The second application explores the specificity of order attribution on the Toronto Stock Exchange, where brokers can choose to trade with their own identity or under a generic anonymous identifier. This leads to two different sources of price impact that degrade the traded asset price both 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. We obtain a semi-explicit solution that helps understand the strategic component of the identity-optionality and the parameter sensitivity of the agents’ optimal trading schedule.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 some existing turnpike estimates for specific classes of Mean Field Games and extend the literature by establishing the result for a linear-quadratic MFG. We build upon the increasingly popular deep-learning algorithm known as the Deep Galerkin Method (DGM) in order to solve MFGs numerically through a “turnpike-accelerated” version of it. The fundamental idea is to include the turnpike estimates into the DGM loss function to obtain solutions more efficiently and with greater accuracy. The performance of the proposed algorithm is demonstrated through a comparative numerical analysis with the baseline DGM algorithm.Overall, this dissertation brings insight, both theoretical and numerical, to rational agents’ optimal behavior in models tailored for each of the above applications.