Abstract
The paper deals with the estimation of parameters of multidimensional diffusion processes that are discretely observed. We construct estimator of the parameters based on the minimum Hellinger distance method. This method is based on the minimization of the Hellinger distance between the density of the invariant distribution of the diffusion process and a nonparametric estimator of this density. We give conditions which ensure the existence of an invariant measure that admits density with respect to the Lebesgue measure and the strong mixing property with exponential rate for the Markov process. Under this condition, we define an estimator of the density based on kernel function and study his properties (almost sure convergence and asymptotic normality). After, using the estimator of the density, we construct the minimum Hellinger distance estimator of the parameters of the diffusion process and establish the almost sure convergence and the asymptotic normality of this estimator. To illustrate the properties of the estimator of the parameters, we apply the method to two examples of multidimensional diffusion processes.
Highlights
Diffusion processes are widely used for modeling purposes in various fields, especially in finance
As a diffusion process is Markovian, the maximum likelihood estimation is the natural choice for parameter estimation to get consistent and asymptotical normally estimator when the transition probability density is known [1]
In the discrete case, for most diffusion processes, the transition probability density is difficult to calculate explicitly which prevents the use of this method
Summary
Diffusion processes are widely used for modeling purposes in various fields, especially in finance. We study the multidimensional diffusion model dXt = a ( Xt ,θ ) dt + b ( Xt ,θ ) dWt , t ≥ 0 under the condition that Xt is positive recurrent and exponentially strong mixing. Using the density of the invariant distribution of the diffusion, we construct an estimator of θ based on minimum Hellinger distance method. The minimum Hellinger distance estimators have been used in parameter estimation for independent observations [7], for nonlinear time series models [8] and recently for univariate diffusion processes [9]. Consistence and asymptotic normality of the kernel estimator of the density of the invariant distribution are studied in the same section.
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