Singular Control, Ambiguity and Real Options

Uncertainty in decision making is a core challenge in economics, finance, and operations research. While classical models address risk with known probabilities, real-world environments often involve ambiguity, where probabilities are unknown or misspecified. Ambiguity arises in corporate cash holdings, climate-driven water management, and inventory control, each structured as an inventory-type problem in which resources accumulate or deplete and interventions occur once thresholds are reached. This thesis develops singular stochastic control models that explicitly incorporate ambiguity, extending the scope of real options theory beyond risk-based formulations.

The thesis progresses from restrictive to more flexible ambiguity frameworks. The first study analyzes cash management under maxmin utility with κ-ignorance, showing via Dynkin games that extreme ambiguity aversion narrows inaction regions and raises expected costs, providing a tractable benchmark. The second study considers reservoir management under smooth ambiguity with a finite-state hidden variable, capturing flood-drought dynamics under climate uncertainty. Using forward-backward stochastic differential equations (FBSDEs), Hamilton-Jacobi-Bellman variational inequalities, and an efficient Markov chain approximation scheme, it shows how ambiguity aversion accelerates interventions while learning gradually mitigates this conservatism. The third study advances to smooth ambiguity with Gaussian hidden variables in a finite-horizon inventory setting, where the problem is formulated through an FBSDE with quadratic growth. This analysis uncovers novel patterns: while higher risk typically delays action, under deep ambiguity, it can instead prompt earlier and more cautious interventions.

Together, these contributions enrich the theory of singular control by embedding ambiguity preferences within nonlinear expected utility, advance analytical and numerical techniques of independent interest, and offer practical insights for managing cash, water, and inventory under deep uncertainty. Conceptually, all models act as forms of ambiguity insurance, quantifying the additional strategic costs required to safeguard decisions in environments where probabilities are themselves uncertain.