Wednesday, September 6, 2023

10:00 a.m. (sharp)

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PhD Candidate: Yichao Chen

Co-Supervisors: Luis Seco, Sebastian Jaimungal

Thesis title: Principal Agent Mean-Field Problems and Multi-Period Compliance Problems With Applications in REC Markets

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Mean field type control problems (MFC) and mean field games (MFGs) are common models to

characterize the asymptotic behavior of a spectrum of interacting agents. One of the successful

application of MFG lies in the study of Renewable Energy Certificate (REC) markets, where there are

interacting firms regulated by a regulator through some payment structures imposed at the end of the

(possibly multiple) compliance periods. The structure of the REC markets motivates us to model it

with principal agent games and multi-period games in a mean-field approach. This thesis investigates a

family of principal agent mean field problems and a family form of multi-period mean filed games with

applications in REC market modelling.

The thesis contains three parts. First we formulate a family of MF-PA problems as an principal’s

optimization problem linked to the terminal conditions of a collection of MV-FBSDE systems. This

fomulation describes the scenario where the principal affect the agents’ equilibria through the agents’

terminal cost. Under suitable assumptions, we proved the well-posedness of the proposed family of

MF-PA problems and we showed the approximation consistency with respect to the principal’s objective

where the MV-FBSDE systems are replaced by its discretized versions. We provide some examples of

PA-MF problems and verify the well-posedness and the approximation consistency. Second, we use

principal agent MFG to model the regulating problem of the REC markets, where the agents form a

Nash equilibria according to the principal’s penalty function, and the principal evaluates the resulting

equilibria. We propose and implement an alternating optimization scheme, based on deep-BSDE

method, to numerically solve the PA-MFG for the REC markets. Our numerical results demonstrate

the efficacy of the algorithm and provide intriguing insights of the REC market regulating modelling

in the mean-field limit. Last but not the least, we introduce a mean field game framework for the

multi-period compliance problem in REC markets. We study a broad family of terminal penalties

generalizing the simple penalty that is proportional to the amount of lacking in the inventory. We argue

for the convexity of the cumulative terminal penalties with respect to the cumulative inventory. Under

suitable regularity condition of the objective function and in both indefinite banking and finite banking

scenarios, we apply variational analysis to the agents’ objective and derive the optimal controls of the

agents together with an explicit equilibrium price that clears the market. We then derive a MV-FBSDE

system that characterizes the equilirbium of the multi-period MFG.

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The draft of the thesis can be found here: Yichao_Chen_Thesis

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