2026 Theses Doctoral

Totally Inaccessible Stopping Times, Path-dependent Volatility and Quadratically Regularized Optimal Transport

Riveros Valdevenito, Andres Ignacio

This dissertation brings together research carried out during my PhD program, in areas ranging from mathematical finance to optimal transport theory. It begins by showing that the Cox construction of totally inaccessible stopping times arises naturally when looking at Markovian jump times, thus supporting its role as a universal tool to study these random times.

This work also covers the existence, uniqueness, positivity and martingality properties of the path-dependent volatility model introduced by Guyon and Lekeufack, providing a thorough theoretical analysis that supports numerical findings. Finally, it characterizes two properties of the quadratically regularized variant of the optimal transport problem.

Firstly, that the empirically observed monotonicity property of the support of the regularized solution, as the regularization parameter shrinks, fails in general. Then, it provides theoretical guarantees for the linear convergence of several algorithms proposed in the literature to approximate the regularized solution. Beginning with implicit results, through a functional-analytic contraction argument, and later refining the analysis for more general setups, by establishing a PL–type inequality, obtaining explicit constants for said linear convergence.