Me, Myself and I: time-inconsistent stochastic control, contract theory and backward stochastic Volterra integral equations
Creators: Hernandez Ramirez, Miguel Camilo
Publisher: name: Columbia University
Year: 2021
Subjects: subject: Mathematics
subject: FOS: Mathematics
subject: Economics
subject: Contracts--Mathematical models
subject: Volterra equations
subject: Stochastic control theory--Statistical methods

description: This thesis studies the decision-making of agents exhibiting time-inconsistent preferences and its implications in the context of contract theory. We take a probabilistic approach to continuous-time non-Markovian time-inconsistent stochastic control problems for sophisticated agents. By introducing a refinement of the notion of equilibrium, an extended dynamic programming principle is established. In turn, this leads to consider an infinite family of BSDEs analogous to the classical Hamilton–Jacobi–Bellman equation. This system is fundamental in the sense that its well-posedness is both necessary and sufficient to characterise equilibria and its associated value function. In addition, under modest assumptions, the existence and uniqueness of a solution is established. With the previous results in mind, we then study a new general class of multidimensional type-I backward stochastic Volterra integral equations. Towards this goal, the well-posedness of a system of an infinite family of standard backward stochastic differential equations is established. Interestingly, its well-posedness is equivalent to that of the type-I backward stochastic Volterra integral equation. This result yields a representation formula in terms of semilinear partial differential equation of Hamilton–Jacobi–Bellman type. In perfect analogy to the theory of backward stochastic differential equations, the case of Lipschitz continuous generators is addressed first and subsequently the quadratic case. In particular, our results show the equivalence of the probabilistic and analytic approaches to time-inconsistent stochastic control problems. Finally, this thesis studies the contracting problem between a standard utility maximiser principal and a sophisticated time-inconsistent agent. We show that the contracting problem faced by the principal can be reformulated as a novel class of control problems exposing the complications of the agent’s preferences. This corresponds to the control of a forward Volterra equation via constrained Volterra type controls. The structure of this problem is inherently related to the representation of the agent’s value function via extended type-I backward stochastic differential equations. Despite the inherent challenges of this class of problems, our reformulation allows us to study the solution for different specifications of preferences for the principal and the agent. This allows us to discuss the qualitative and methodological implications of our results in the context of contract theory: (i) from a methodological point of view, unlike in the time-consistent case, the solution to the moral hazard problem does not reduce, in general, to a standard stochastic control problem; (ii) our analysis shows that slight deviations of seminal models in contracting theory seem to challenge the virtues attributed to linear contracts and suggests that such contracts would typically cease to be optimal in general for time-inconsistent agents; (iii) in line with some recent developments in the time-consistent literature, we find that the optimal contract in the time-inconsistent scenario is, in general, non-Markovian in the state process X.