Backward SDEs and Applications to Optimal Control Problems

Abstract

In this thesis, we delve into two distinct facets, one theoretical and the other practical. The theoretical aspect of our investigation centers on the examination of backward stochastic differential equations driven by both a Poisson process and an independent Brownian motion succinctly denoted as BSDEJs. The generator showcases logarithmic growth in both the state variable and the process z while retaining Lipschitz continuity concerning the jump component. Our study systematically establishes the presence and distinctiveness of solutions within appropriate functional spaces. Furthermore, we loosen the Lipschitz condition on the Poisson component, allowing the generator to manifest logarithmic growth concerning all variables. Taking an additional stride, we utilize an exponential transformation to draw a parallel between solutions of a BSDEJ characterized by quadratic growth in the z-variable and a BSDEJ exhibiting logarithmic growth with both y and z. Additionally, we delve into a discussion on the maximum principle, specifically in scenarios devoid of the jump component. On the practical side, our focus shifts to the implementation of Public-Private Partnerships (PPPs), which have emerged as a promising approach for efficiently managing public infrastructure projects and services. However, the success of PPP contracts is often hindered by challenges such as information asymmetry and moral hazard. To optimize decision-making in PPPs, this thesis focuses on the application of stochastic control techniques, taking into account the effect of the ambiguity factor κ in the contract between the principal and the agent. By leveraging rigorous mathematical frameworks, including one-dimensional BSDEs, techniques in stochastic control, and optimizing stopping times, this research provides valuable insights and practical solutions to mitigate the adverse effects of information asymmetry, ambiguity, and continuous-time dynamics in PPPs. This study derives the HJB Variational Inequality (HJBVI) associated with the public value function, offering a solid foundation for decision-making optimization in PPPs. Additionally, this work conducts a numerical study using finite difference methods and the Howard algorithm to approximate the optimal rent and effort under uncertainty. The numerical analysis demonstrates the impact of uncertainty on decision-making and project outcomes in PPP contracts. Overall, this thesis significantly contributes to the theoretical and applied fields. Firstly, we establish the existence and uniqueness of BSDEJs with a generator allowing for logarithmic growth. Furthermore, we explore the connection of these BSDEJs with quadratic BSDEJs. Secondly, we delve into the Pontryagin maximum principle for these types of BSDEs, specifically without the jump component. Finally, we advance the field of Public-Private Partnerships (PPPs) by optimizing decision-making.