Path-dependent equations and viscosity solutions in infinite dimension

Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus (see [15]), in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [15, 9]) and viscosity solutions (see e.g. [16]). In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.Comment: To appear in the Annals of Probabilit

Path-dependent Hamilton-Jacobi-Bellman equation: Uniqueness of Crandall-Lions viscosity solutions

We prove existence and uniqueness of Crandall-Lions viscosity solutions of Hamilton-Jacobi-Bellman equations in the space of continuous paths, associated to the optimal control of path-dependent SDEs. This seems the first uniqueness result in such a context. More precisely, similarly to the seminal paper of P.L. Lions, the proof of our core result, that is the comparison theorem, is based on the fact that the value function is bigger than any viscosity subsolution and smaller than any viscosity supersolution. Such a result, coupled with the proof that the value function is a viscosity solution (based on the dynamic programming principle, which we prove), implies that the value function is the unique viscosity solution to the Hamilton-Jacobi-Bellman equation. The proof of the comparison theorem in P.L. Lions' paper, relies on regularity results which are missing in the present infinite-dimensional context, as well as on the local compactness of the finite-dimensional underlying space. We overcome such non-trivial technical difficulties introducing a suitable approximating procedure and a smooth gauge-type function, which allows to generate maxima and minima through an appropriate version of the Borwein-Preiss generalization of Ekeland's variational principle on the space of continuous paths

Topics in stochastic calculus in infinite dimension for financial applications

This thesis is devoted to study delay/path-dependent stochastic differential equations and their connection with partial differential equations in infinite dimensional spaces, possibly path-dependent. We address mathematical problems arising in hedging a derivative product for which the volatility of the underlying assets as well as the claim may depend on the past history of the assets themselves. The starting point is to provide a robust framework for working with mild solutions to path-dependent SDEs: well-posedness, continuity with respect to the data, regularity with respect to the initial condition. This is done in Chapter 1. In Chapter 2, under Lipschitz conditions on the data, we prove the directional regularity needed in order to write the hedging strategy. In Chapter 3 we introduce a new notion of viscosity solution to semilinear path-dependent PDEs in Hilbert spaces (PPDEs), we prove well-posedness and show that the solution is given by the Fyenman-Kac formula. In Chapter 4 we extend to Hilbert spaces the functional It\uafo calculus and, under smooth assumptions on the data, we prove a path-dependent It\uafo\u2019s formula, show existence of classical solutions to PPDEs, and obtain a Clark-Ocone type formula. In Chapter 5 we introduce a new notion of C0-semigroup suitable to be applied to Markov transition semigroups, hence to mild solutions to Kolmogorov PDEs, and we prove all the basic results analogous to those available for C0-semigroups in Banach spaces. Additional theoretical results for stochastic analysis in Hilbert spaces, regarding stochastic convolutions, are given in Appendix A. Our methodology varies among different chapters. Path-dependent models can be studied in their original path-dependent form or by representing them as non-pathdependent models in infinite dimension. We exploit both approaches. We treat pathdependent Kolmogorov equations in infinite dimension with two notions of solution: classical and viscosity solutions. Each approach leads to original results in each chapter

Il cambiamento climatico: un'analisi economica

ope

Análise geomorfológica aplicada ao saneamento básico, no perímetro urbano de Cáceres - Mato Grosso

Abstrac

Histórias visuais de partos: vídeos de afeto e informação

Histórias visuais de partos: vídeos de afeto e informação consiste em um Trabalho de Conclusão de Curso que visa tornar acessível a informação sobre as vias de parto, bem como incentivar o registro do mesmo em foto e vídeo, como forma de eternizar em memórias esse importante momento. Ele tem como público-alvo gestantes que planejam ter seus bebês em hospitais particulares do Rio de Janeiro. Como ponto de partida, estudamos o fenômeno do nascimento, os tipos de parto mais comuns e os primeiros sinais do trabalho de parto. Além disso, buscamos dados sobre os nascimentos no Brasil e constatamos que há uma maior proporção de partos através do procedimento cirúrgico, o que traz um alerta para a falta de informação. Em seguida, discorremos sobre como documentar histórias com imagens, percorrendo desde o surgimento da fotografia até o seu desdobramento na fotografia documental. Também, discutimos referências de imagens de parto no Campo da Arte, a fim de entender a representação da temática ao longo do tempo. Ainda, entrevistamos mães e profissionais da fotografia para aprofundar e entender suas experiências individuais quanto ao parto e quanto à relevância do registro imagético nesse momento. Por fim, apresentamos o desenvolvimento do projeto em si: os vídeos, a escrita dos roteiros textuais e imagéticos e edição das imagens e tipografia. Os vídeos propostos estão separados pelas temáticas "Preparação para o parto", "O dia do parto" e "O valor do registro após o parto"