Variable bandwidth local maximum likelihood type estimation for diffusion processes

Ming T Tang,Yun Y Wang

doi:10.1186/s13662-018-1506-1

Abstract

The method of robust approach is applied to estimate drift function and diffusion function of diffusion processes with discrete-time observations. The proposed method combines the ideas of local linear regression technique and maximum likelihood type estimation technique, so the advantages of local linear estimators persist and overcome the disadvantages of least-squares estimator. Moreover, a variable bandwidth instead of a constant bandwidth is considered in the local maximum likelihood type estimators. The consistency and asymptotic normality of the local maximum likelihood type estimators for drift and diffusion functions are developed under some given conditions. We perform a simulation study to evaluate the robust performances of the proposed estimators.

Highlights

Diffusion processes X defined by the following stochastic differential equation are considered in this article: dXt = μ(Xt) dt + σ (Xt) dBt, (1)where {Bt, t ≥ 0} is a standard Brownian motion, μ(·) is an unknown measurable function and σ (·) is an unknown positive function
Recently in the literature, the statistical inference for diffusion processes based on discrete observations has often been of concern; for example, see [3,4,5,6] and its references for parametric estimation, see [7,8,9] and the references
The purpose of this paper is to investigate the local linear and variable bandwidth Mestimators of the drift and diffusion functions in model (1) based on high-frequency data, that is, the sample observations are only selected at discrete-time points, say at n spaced {i, i = 0, 1, . . . , n}, where is the sampling interval, and → 0 as n → ∞

Summary

Introduction

Reference [2] improved estimation of drift parameters of diffusion processes for interest rates by incorporating information in bond prices. Recently in the literature, the statistical inference for diffusion processes based on discrete observations has often been of concern; for example, see [3,4,5,6] and its references for parametric estimation, see [7,8,9] and the references. The first to consider nonparametric estimation for the diffusion coefficient in model (1) with discrete-time observation was [17], where a kernel type estimator was considered. Reference [12] generalized Stanton’s idea and introduced the local polynomial estimators for drift and diffusion functions. Since a local linear method may produce negative values for the diffusion function, [14] proposed a new nonparametric estimation procedure of the diffusion function based on re-weighting the Nadaraya–Watson estimator

Objectives

Conclusion