Corrigendum to "Weak Approximations for Wiener Functionals" [Ann. Appl. Probab. (2013), 23, 4, 1660-1691

The proofs of Theorem 3.1 and Corollary 4.1 in Le\~ao and Ohashi (2013) are incomplete. The reason is a wrong statement in Remark 2.2. The hypotheses and statements of Theorem 3.1 and Corollary 4.1 in Le\~ao and Ohashi (2013) remain unchanged but the proofs have to be modified. In this short note, we provide the details.Comment: Errata of the paper Weak approximations for Wiener functional

Discrete-type approximations for non-Markovian optimal stopping problems: Part I

In this paper, we present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\epsilon$-optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent SDEs driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte-Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra, Ohashi and Russo.Comment: Final version to appear in Journal of Applied Probabilit

Stochastic Near-Optimal Controls for Path-Dependent Systems

In this article, we present a general methodology for control problems driven by the Brownian motion filtration including non-Markovian and non-semimartingale state processes controlled by mutually singular measures. The main result of this paper is the development of a concrete pathwise method for characterizing and computing near-optimal controls for abstract controlled Wiener functionals. The theory does not require ad hoc functional differentiability assumptions on the value process and elipticity conditions on the diffusion components. The analysis is pathwise over suitable finite dimensional spaces and it is based on the weak differential structure introduced by Le\~ao, Ohashi and Simas jointly with measurable selection arguments. The theory is applied to stochastic control problems based on path-dependent SDEs where both drift and possibly degenerated diffusion components are controlled. Optimal control of drifts for path-dependent SDEs driven by fractional Brownian motion is also discussed. We finally provide an application in the context of financial mathematics. Namely, we construct near-optimal controls in a non-Markovian portfolio optimization problem.Comment: We shorten some of the proofs, the Introduction was updated and a concrete example to Mathematical Finance is presente

A general Multidimensional Monte Carlo Approach for Dynamic Hedging under stochastic volatility

In this work, we introduce a Monte Carlo method for the dynamic hedging of general European-type contingent claims in a multidimensional Brownian arbitrage-free market. Based on bounded variation martingale approximations for Galtchouk-Kunita-Watanabe decompositions, we propose a feasible and constructive methodology which allows us to compute pure hedging strategies w.r.t arbitrary square-integrable claims in incomplete markets. In particular, the methodology can be applied to quadratic hedging-type strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. We illustrate the method with numerical examples based on generalized Follmer-Schweizer decompositions, locally-risk minimizing and mean-variance hedging strategies for vanilla and path-dependent options written on local volatility and stochastic volatility models.Comment: Some typos are corrected in Section

A weak version of path-dependent functional It\^o calculus

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the "sensitivities" of processes, namely derivatives of martingale components and a weak notion of infinitesimal generators, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t Brownian motion driving noise.Comment: Version to appear in Annals of Probabilit

Weak differentiability of Wiener functionals and occupation times

In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Le\~ao, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian paths under finite $p$-variation ($p\ge 2$) regularity is also discussed. Under stronger regularity conditions in the sense of finite $(p,q)$-variation, the connection between weak differentiability and two-parameter local time integrals in the sense of Young is established.Comment: Revised version. To appear in Bulletin des Sciences Math\'ematiques. arXiv admin note: text overlap with arXiv:1707.04972, arXiv:1408.142

Weak Functional It\^o Calculus and Applications

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration. The theory provides a differential structure which describes the infinitesimal evolution of Wiener functionals at very small scales. The Markov property is replaced by a finite-dimensional approximation procedure based on controlled inter-arrival times and jumps of approximating martingales. The theory reveals that a large class of adapted processes follow a differential rule which is similar in nature to a fundamental theorem of calculus in the context of Wiener functionals. Null stochastic derivative term turns out to be a non-Markovian version of the classical second order parabolic operator and connections with theory of local-times and $(p,q)$-variation regularity are established. The framework extends the pathwise functional calculus and it opens the way to obtain variational principles for processes. Applications to semi-linear variational equations and stochastic variational inequalities are presented. In particular, we provide a feasible dynamic programming principle for fully non-Markovian/non-semimartingale optimal stopping problems.Comment: The results of the article were reformulated and split into two new paper

On the Discrete Cram\'er-von Mises Statistics under Random Censorship

In this work, nonparametric log-rank-type statistical tests are introduced in order to verify homogeneity of purely discrete variables subject to arbitrary right-censoring for infinitely many categories. In particular, the Cram\'er-von Mises test statistics for discrete models under censoring is established. In order to introduce the test, we develop the weighted log-rank statistics in a general multivariate discrete setup which complements previous fundamental results of Gill (1980) and Andersen et al. (1982). Due to the presence of persistent jumps over the unbounded set of categories, the asymptotic distribution of the test is not distribution-free. The statistical test for a large class of weighted processes is described as a weighted series of independent chi-squared variables whose weights can be consistently estimated and the associated limiting covariance operator can be infinite-dimensional. The test is consistent to any alternative hypothesis and, in particular, it allows us to deal with crossing hazard functions. We also provide a simulation study in order to illustrate the theoretical results.Comment: 50 page

Estimation of Distribution Algorithm for Protein Structure Prediction

Proteins are essential for maintaining life. For example, knowing the structure of a protein, cell regulatory mechanisms of organisms can be modeled, supporting the development of disease treatments or the understanding of relationships between protein structures and food attributes. However, discovering the structure of a protein can be a difficult and expensive task, since it is hard to explore the large search to predict even a small protein. Template-based methods (coarse-grained, homology, threading etc) depend on Prior Knowledge (PK) of proteins determined using other methods as X-Ray Crystallography or Nuclear Magnetic Resonance. On the other hand, template-free methods (full-atom and ab initio) rely on atoms physical-chemical properties to predict protein structures. In comparison with other approaches, the Estimation of Distribution Algorithms (EDAs) can require significant less PK, suggesting that it could be adequate for proteins of low-level of PK. Finding an EDA able to handle both prediction quality and computational time is a difficult task, since they are strong inversely correlated. We developed an EDA specific for the ab initio Protein Structure Prediction (PSP) problem using full-atom representation. We developed one univariate and two bivariate probabilistic models in order to design a proper EDA for PSP. The bivariate models make relationships between dihedral angles $\phi$ and $\psi$ within an amino acid. Furthermore, we compared the proposed EDA with other approaches from the literature. We noticed that even a relatively simple algorithm such as Random Walk can find the correct solution, but it would require a large amount of prior knowledge (biased prediction). On the other hand, our EDA was able to correctly predict with no prior knowledge at all, characterizing such a prediction as pure ab initio.Comment: 45 pages, 13 figure

Data Analysis for Proficiency Testing

Proficiency Testing (PT) determines the performance of individual laboratories for specific tests or measurements and it is used to monitor the reliability of laboratories measurements. PT plays a highly valuable role as it provides an objective evidence of the competence of the participant laboratories. In this paper, we propose a multivariate model to assess equivalence among laboratories measurements in proficiency testing. Our method allow to include type B source of variation and to deal with multivariate data, where the item under test is measured at different levels. Although intuitive, the proposed model is nonergodic, which means that the asymptotic Fisher information matrix is random. As a consequence, a detailed asymptotic analysis was carried out to establish the strategy for comparing the results of the participating laboratories. To illustrate, we apply our method to analyze the data from the Brazilian Engine test group, PT program, where the power of an engine was measured by 8 laboratories at several levels of rotation