Parameter Estimation for Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion

Na Song,Zaiming Liu

doi:10.1155/2014/942307

Abstract

We study the asymptotic properties of minimum distance estimator of drift parameter for a class of nonlinear scalar stochastic differential equations driven by mixed fractional Brownian motion. The consistency and limit distribution of this estimator are established as the diffusion coefficient tends to zero under some regularity conditions.

Highlights

Stochastic differential equations (SDEs) are a natural choice to model the time evolution of dynamic systems which are subject to random influences
Kouame et al [19] studied asymptotic properties of minimum distance estimator of the parameter of stochastic process driven by a fBm as the diffusion coefficient tends to zero
It appears that there are few works studying the estimators of mfBm

Summary

Introduction

Stochastic differential equations (SDEs) are a natural choice to model the time evolution of dynamic systems which are subject to random influences. Kouame et al [19] studied asymptotic properties of minimum distance estimator of the parameter of stochastic process driven by a fBm as the diffusion coefficient tends to zero. It appears that there are few works studying the estimators of mfBm. Zili [20] obtained some general stochastic properties of the mfBm and treated the Holder continuity of the sample paths and α-differentiability of the trajectories of mfBm. Miao [21] obtained the asymptotic properties of the minimum L1-norm estimator of the drift parameter for a linear SDE driven by an mfBm. Xiao et al [22] studied the problem of estimating the parameters for the mfBm from discrete observations based on the MLE.

Notation and Preliminaries

Consistency

Asymptotic Distribution