Coleção Insper Business and Economics Working Papers
URI permanente para esta coleçãohttps://repositorio.insper.edu.br/handle/11224/5740
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Working Paper Weak Approximations for Wiener Functionals(2012) Leão, Dorival; Ohashi, Alberto Masayoshi FariaIn this paper we introduce a simple space-filtration discretization scheme on Wiener space which allows us to study weak decompositions and smooth explicit approximations for a large class of Wiener functionals. We show that any Wiener functional has an underlying robust semimartingale skeleton which under mild conditions converges to it. The discretization is given in terms of discrete-jumping filtrations which allow us to approximate non-smooth processes by means of a stochastic derivative operator on the Wiener space. As a by-product, we provide a robust semimartingale approxi mation for weak Dirichlet-type processes. The underlying semimartingale skeleton is intrinsically constructed in such way that all the relevant structure is amenable to a robust numerical scheme. In order to illustrate the results, we provide an easily implementable approxi mation scheme for the classical Clark-Ocone formula in full generality. Unlike in previous works, our methodology does not assume an underlying Markovian structure and does not require Malliavin weights. We conclude by proposing a method that enables us to compute optimal stopping times for possibly non Markovian systems arising e.g. from the fractional Brownian motion.Working Paper On the Discrete Cramér-von Mises Statistics under Random Censorship(2012) Leão, Dorival; Ohashi, Alberto Masayoshi FariaIn 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 [13] and Andersen et al. [5]. 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. Moreover, the associated limiting covariance operator can be infinite-dimensional which allows us to deal consistently with an infinite survival time typically founded in long-term survival analysis such as cure-rate models. 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.Working Paper Weak Approximations for Wiener Functionals(2010) Leão, Dorival; Ohashi, Alberto Masayoshi FariaIn this paper we introduce a simple space-filtration discretization scheme on Wiener space which allows us to study weak decompositions and smooth approximations for a large class of Wiener functionals. We show that any Wiener functional has an underlying robust semimartingale skeleton which under mild conditions converges to it. The approximation is given in terms of discrete-jumping filtrations which allow us to approximate irregular processes by means of a stochastic derivative operator on Wiener space. As a by-product, we prove that continuous paths and a suitable notion of energy are sufficient in order to get a unique orthogonal decomposition similar to weak Dirichlet processes. In this direction, we generalize the main results given in Graversen and Rao [29] and Coquet et al. [12] in the particular Brownian filtration case. The second part of this paper is devoted to the application of these abstract results to concrete non-smooth processes. We show that our embedded semimartingale structure provides an easily implementable approximation scheme for the classical Clark-Ocone formula in full generality. Unlike in previous works our methodology does not assume an underlying Markovian structure and requires no use of Malliavin weights as in the classical literature of Mathematical Finance.