Optimal Stopping of Gauss-Markov processes

Mención Internacional en el título de doctorIn this thesis we contribute to the optimal stopping theory literature, in the time-inhomogeneous framework, by solving Optimal Stopping Problems (OSPs) related to Gauss–Markov (GM) processes, both when they are non-degenerated, and when they are pinned to a deterministic value at a terminal time. For pinned processes, we bypassed the challenge of their explosive drifts by equating them to a time-space-transformed Brownian Motion (BM). For each OSP, we characterized the free-boundary equation as the unique solution of a type-two Volterra integral equation. The value functions were, then, expressed as an integral of the OSBs. We used a solution methodology in the spirit of Peskir (2005). That is, a direct, probabilistic approach that harvests sufficient smoothness of the value function and the Optimal Stopping Boundary (OSB) to solve the associated free-boundary problem by using an extended Itô’s lemma. In doing so, we proved the Lipschitz continuity of the OSB away from the horizon. This result extends the technique in De Angelis and Stabile (2019) and blueprints a methodology to obtain similar smoothness on other OSPs. Another highly customizable technique was the one we employed to obtain the OSB’s boundedness. By comparing the non-degenerated GM process and the Gauss Markov Bridge (GMB) with a BM and a Brownian Bridge (BB), respectively, we found bounds for the OSBs of the former two processes from those of the latter two. Two different fixed-point algorithms were presented and implemented to solve the freeboundary equation. One based on backward induction (see Section 3.4) and one based on the Picard iteration method (see Sections 2.5, 4.6, and 5.6). With the aid of these algorithms, we illustrated the geometry of the OSB for different forms of the processes’ drift and volatility (see Figures 2.1, 3.1, 4.1–4.3, and 5.2). It is worth mentioning the statistical inference study we perform on the OSB in the BB case (see Section 3.4), as this is not a typical subject addressed in optimal stopping theory, and it is potentially extensible to tackle more general settings where likelihood theory is worked out. Indeed, the methodology consists in using the asymptotic normality of the BB volatility’s maximum-likelihood estimate to extend, by using the delta method, such property to the OSB plugin estimator. This allowed us to provide (point-wise) confidence curves for the OSB. We also offer a financial perspective of our work in Chapters 2 and 3, by linking the OSPs to the problem of optimally exercising American options. Remarkably, in Section 3.5, we show the competitiveness of the BB model against the geometric BM in this regard, when the option is written on IBM’s and Apple’s stocks, and in the presence of the pinning-at-the-strike effect. In addition, the confidence curves computed in Section 3.4 provide traders with a mechanism to introduce a risk-preference element.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Franciso de Asís Torres Ruiz.- Secretaria: Rosa Elvira Lillo Rodríguez.- Vocal: Tiziano De Angeli

The short memory limit for long time statistics in a stochastic Coleman-Gurtin model of heat conduction

We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials and sufficiently regular noise the system always admits invariant probability measures, defined on the extended phase space, that possess higher regularity properties dictated by the structure of the nonlinearities in the equation. Furthermore, we investigate the singular limit as the memory kernel collapses to a Dirac function. Specifically, provided sufficiently many directions in the phase space are stochastically forced, we show that there is a unique stationary measure to which the system converges, in a suitable Wasserstein distance, at exponential rates independent of the decay of the memory kernel. We then prove the convergence of the statistically steady states to the unique invariant probability of the classical stochastic reaction-diffusion equation in the desired singular limit. As a consequence, we establish the validity of the small memory approximation for solutions on the infinite time horizon $[0,\infty)$

Gauge Theories of Gravitation

During the last five decades, gravity, as one of the fundamental forces of nature, has been formulated as a gauge theory of the Weyl-Cartan-Yang-Mills type. The present text offers commentaries on the articles from the most prominent proponents of the theory. In the early 1960s, the gauge idea was successfully applied to the Poincar\'e group of spacetime symmetries and to the related conserved energy-momentum and angular momentum currents. The resulting theory, the Poincar\'e gauge theory, encompasses Einstein's general relativity as well as the teleparallel theory of gravity as subcases. The spacetime structure is enriched by Cartan's torsion, and the new theory can accommodate fermionic matter and its spin in a perfectly natural way. This guided tour starts from special relativity and leads, in its first part, to general relativity and its gauge type extensions \`a la Weyl and Cartan. Subsequent stopping points are the theories of Yang-Mills and Utiyama and, as a particular vantage point, the theory of Sciama and Kibble. Later, the Poincar\'e gauge theory and its generalizations are explored and special topics, such as its Hamiltonian formulation and exact solutions, are studied. This guide to the literature on classical gauge theories of gravity is intended to be a stimulating introduction to the subject.Comment: 169 pages, pdf file, v3: extended to a guide to the literature on classical gauge theories of gravit

Singular stochastic integral operators

In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a H\"ormander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the $A_2$-conjecture. The results are applied to obtain $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on $\mathbb{R}^d$ and smooth and angular domains.Comment: typos corrected. Accepted for publication in Analysis & PD