A Higher order and stable method for the numerical integration of Random Differential Equations

Trabalho apresentado no Congresso Nacional de Matemática Aplicada à Indústria, 18 a 21 de novembro de 2014, Caldas Novas - Goiás

Exploratory semiparametric analysis of two-dimensional diffusions in finance

We examine bivariate extensions of Aït-Sahalia’s approach to the estimation of univariate diffusions. Our message is that extending his idea to a bivariate setting is not straightforward. In higher dimensions, as opposed to the univariate case, the elements of the Itô and Fokker-Planck representations do not coincide; and, even imposing sensible assumptions on the marginal drifts and volatilities is not sufficient to obtain direct generalisations. We develop exploratory estimation and testing procedures, by parametrizing the drifts of both component processes and setting restrictions on the terms of either the Itô or the Fokker-Planck covariance matrices. This may lead to highly nonlinear ordinary differential equations, where the definition of boundary conditions is crucial. For the methods developed, the Fokker-Planck representation seems more tractable than the Itô’s. Questions for further research include the design of regularity conditions on the time series dependence in the data, the kernels actually used and the bandwidths, to obtain asymptotic properties for the estimators proposed. A particular case seems promising: “causal bivariate models” in which only one of the diffusions contributes to the volatility of the other. Hedging strategies which estimate separately the univariate diffusions at stake may thus be improved.

Wealth composition, endogenous fertility and the dynamics of income inequality

This paper analyzes how differences in the composition of wealth between human and physical capital among families affect fertility choices. These in tum influence the dynamics of wealth and income inequality across generations through a tradeoffbetween quantity and quality of children. Wealth composition affects fertility because physical capital has only a wealth effect on number of children, whereas human capital increases the time cost of child-rearing in addition to the wealth effect. I construct a model combining endogenous fertility with borrowing constraints in human capital investments, in which weaIth composition is determined endogenously. The model is calibrated to the PNAD, a Brazilian household survey, and the main findings of the paper can be summarized as follows. First, the model implies that the crosssection relationship between fertility and wealth typically displays a U-shaped pattem, reflecting differences in wealth composition between poor and rich families. Also, the quantity-quality tradeoff implies a concave cross-section relationship between investments per child and wealth. Second, as the economy develops and families overcome their bOlTowing constraints, the negative effect of weaIth on fertility becomes smaller, and persistence of inequality declines accordingly. The empirical evidence presented in this paper is consistent with both implications .