Nonparametric Estimation of Second-Order Stochastic Differential Equations

Abstract

We propose nonparametric estimators of the infinitesimal coefficients associated with second-order stochastic differential equations+ We show that under appropriate conditions, the proposed estimators are consistent+ Also, we state conditions ensuring the asymptotic normality of these estimators+ We conclude our paper with a Monte Carlo experiment in which we assess the response of the nonparametric estimators with respect to the step of discretization+ I thank two anonymous referees who made valuable suggestions that led to considerable improvements in the paper+ I am also grateful to Carlos Braumann and Tom Kundert for helpful comments+ This research was supported by the Fundação para a Ciência e a Tecnologia~FCT! and by FEDER0POCI 2010+ Address correspondence to João Nicolau,

Modeling financial time series through second-order stochastic differential equations

“…We provide an empirical illustration based on non-parametric estimators discussed in Nicolau (2007). The estimation results suggest a parametric second order stochastic di¤erential equation for stock prices and exchange rates, which we brie ‡y discussed.…”

“…Non-parametric estimation of integrated di¤usions based on the observations nX i o is analyzed in Nicolau (2007). Gloter (1999Gloter ( , 2006 and Ditlevsen and Sørensen (2004) analyze parametric and semi-parametric estimation.…”

“…Under some conditions (including stationarity of X; and that the step of discretization goes to zero at an appropriate rate) Nicolau (2007) has shown thatâ…”

“…performs better thanâ n (x) for moderate/high values of (see Nicolau, 2007). We use a n (x) and b 2 n (x) to estimate the in…nitesimal coe¢ cients a (x) and b 2 (x) associated with these nine …nancial time series (X i is given by equation (4) and Y is the log of price).…”

Nonparametric robust function estimation for integrated diffusion processes

“…For example, parametric inference for integrated diffusion processes has been addressed by [11,15,16,17]. As for the nonparametric inferences, [22] obtained the Nadaraya-Waston estimators for drift and diffusion functions, [24] studied the local linear estimators for the two functions, and [25] developed the re-weighted estimator of the diffusion function.…”