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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.

Diffusion processes are widely used for modeling purposes in various fields, especially in finance. Many papers are devoted to the parameter estimation of the drift and diffusion coefficients of diffusion processes by discrete observation. 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 [

In this paper, we study the multidimensional diffusion model

under the condition that

Let

The interest for this method of parametric estimation is that the minimum Hellinger distance estimation method gives efficient and robust estimators [

The paper is organized as follows. In Section 2, we present the statistical model and some conditions which imply that

We consider the d-dimensional diffusion process solution of the multivariate stochastic differential equation:

where

We assume that the functions a and b are known up to the parameter

We denote by

For a matrix

The process

We make the following assumptions on the model:

(A_{1}): there exists a constant C such that

(A_{2}): there exist constants

(A_{3}): the matrix function

Assumptions (A_{1})-(A_{3}) ensure the existence of a unique strong solution for the Equation (1) and an invariant measure for the process

In the sequel, we assume that the initial value

We consider the kernel estimator

where

(A_{4})

(1) There exists

(2)

(A_{5})

We finish with assumptions concerning the density of the invariant distribution:

(A_{6})

(A_{7})

Properties (consistence and asymptotic normality) of the kernel density estimator are examined in the following theorems. The proof of the two theorems can be found in the Appendix.

Theorem 1. Under assumptions (A_{1})-(A_{4}), if the function

Theorem 2. Under assumptions (A_{1})-(A_{6}), if

distribution of

The minimum Hellinger distance estimator of

where

Let

Define the functional

where

If

Theorem 3. (almost sure consistency)

Assume that assumptions (A_{1})-(A_{4}) and (A_{7}) hold. If for all

Proof. By Theorem 1,

Using the inequality

Since

By theorem 1 [

ger topology. Therefore

This achieves the proof of the theorem.

Denote

when these quantities exist. Furthermore, let

To prove asymptotic normality of the estimator of the parameter, we begin with two lemmas.

Lemma 1. Let

(1) assumptions (A_{1})-(A_{5}) are satisfied,

(2)

then for any positive sequence

The proof can be found in the Appendix.

Remark 1. The two dimensional stochastic process (see Section 4) with invariant density

where

Lemma 2. Let _{1})-(A_{6}) are satisfied and:

(2)

then

The proof can be found in the Appendix.

Remark 2. Let

numbers diverging to infinity. Let

where

Theorem 4. (asymptotic normality)

Under assumption (A_{7}) and conditions of Lemma 1 and Lemma 2, if

(1) for all

(2) the components of

(3)

Proof. From Theorem 2 [

where

We have

Denote

We have

where

By Lemma 2,

since

Therefore the limiting distribution of

where

This completes the proof of the theorem.

We consider the two-dimensional Ornstein-Uhlenbeck process solution of the stochastic differential equation

where

Let

_{1})-(A_{3}). Therefore,

Furthermore [

The solution of the Equation (3) is

Therefore [

The minimum Hellinger distance estimator of

where

with

where _{4}) and (A_{5}) such that

Let

which gives the the following system

Thus,

We now give simulations for different parameter values using the R language. For each process, we generate sample paths using the package “sde” [

bandwidth

Simulations are based on 1000 observations of the Ornstein-Uhlenbeck process with 200 replications.

Simulation results are given in the

Means | Standard errors | |

(0.3, 0.7) | (0.2985977, 0.6998527) | (0.01076013, 0.01032311) |

(0.5, 2) | (0.4954066, 1.998997) | (0.0341282, 0.008429909) |

(1, 2.4) | (0.9987882, 2.398991) | (0.009604874, 0.01262858) |

(1, 3) | (0.998918, 2.999193) | (0.008726621, 0.01034987) |

(0.223, 0.6) | (0.2223449, 0.6006928) | (0.01048311, 0.01224315) |

In

We consider the Homogeneous Gaussian diffusion process [

where

Let

As in [

Then we have

Let

Let

True values of | ||||
---|---|---|---|---|

Means | Standard errors | Means | Standard errors | |

3.996942 | 0.0005203 | 4.0349 | 0.2904 | |

1.00776 | 0.001311968 | 1.0035 | 0.2891 | |

−2.007696 | 0.001315799 | −2.0155 | 0.1248 | |

−2.982749 | 0.002923666 | −3.0247 | 0.1978 | |

1.009081 | 0.001513984 | 1.0078 | 0.1177 |

For simulation, we must write the stochastic differential Equation (4) in matrix form as follows:

As in [

Now, we can simulate a sample path of the Homogeneous Gaussian diffusion using the “yuima” package of R language [

We generate 500 sample paths of the process, each of size 500. The kernel function and the bandwidth are those of the previous example.

We compare the estimator obtained by the minimum Hellinger distance method (MHD) of this paper and the estimator obtained in [

Julien ApalaN’drin,OuagninaHili, (2015) From Nonparametric Density Estimation to Parametric Estimation of Multidimensional Diffusion Processes. Applied Mathematics,06,1592-1610. doi: 10.4236/am.2015.69142

Proof.

We have:

Step 1:

by Theorem 2.1 [

Hence

Step 2:

where

Then by theorem 2.1 [

We have

where

Then

Therefore

by the Borel-Cantelli’s lemma.

(6) and (7) imply that

This achieves the proof of the theorem.

A2. Proof of Theorem 2Proof.

(1)

By making the change of variable _{4}) and (A_{5}), we get:

(2)

where

We have

Let

Define

and

We have

Step 1: We prove that

By Minkowski’s inequality, we have

(1) Using Billingsley’s inequality [

(2)

Hence,

Therefore, choosing

we get

which implies that

Step 2: asymptotic normality of

From Lemma 4.2 [

Setting

the charasteristic function of

We have

(2) Note that

Therefore

Since the random variables

the limiting distribution of

The condition (8), (9) and (10) are satisfied, for example, with

This achieves the proof of the theorem.

A3. Proof of Lemma 1Proof. The proof of the lemma is done in two steps.

Step 1: we prove that

With assumptions (A_{4}) and (A_{5}), we have

Furthermore,

Therefore

Step 2: asymptotic normality of

Proof is similar to that of theorem 2; we use the inequality of Davidov [

Note that:

and

(2)

Recall that

Let

strongly mixing with mean zero and variance

From (1),

Therefore,

This completes the proof of the lemma.

A4. Proof of Lemma 2Proof.

We have,

Now,

(1)

Using Davidov’s inequality for mixing processes, we get

Choose

Hence,

(2)

Therefore,

The last relation implies that

Furthermore,

We have,

Therefore, if

then

(11) and (12) imply that