Martingale estimation functions for Bessel processes

In this paper we derive martingale estimating functions for the dimensionality parameter of a Bessel process based on the eigenfunctions of the diffusion operator. Since a Bessel process is non-ergodic and the theory of martingale estimating functions is developed for ergodic diffusions, we use the space-time transformation of the Bessel process and formulate our results for a modified Bessel process. We deduce consistency, asymptotic normality and discuss optimality. It turns out that the martingale estimating function based of the first eigenfunction of the modified Bessel process coincides with the linear martingale estimating function for the Cox Ingersoll Ross process. Furthermore, our results may also be applied to estimating the multiplicity parameter of a one-dimensional Dunkl process.


Introduction
Martingale estimating functions introduced in [bs1995] provide a well-established method for inference in discretely observed diffusion processes, when the likelihood function is unknown or too complicated. The idea behind martingale estimating functions is to provide a simple approximation of the true likelihood, which forms a martingale and hence leads under suitable regularity assumptions to consistent and asymptotically normal estimators. One way of approximating the likelihood function is by Taylor expansion leading to linear and quadratic martingale estimating functions, cf. [bs1995]. Another possibility is to use the eigenfunctions of the associated diffusion operator, cf. [ks1999]. In this context a suitable optimality concept was introduced by [gh1987] and [h1988]. For a general theory of asymptotic statistics for diffusion processes we refer e.g. to [h2014].
Our aim in this paper is to estimate the dimensionality or index parameter ϑ ∈ Θ ⊂ (− 1 2 , ∞) of a classical one-dimensional Bessel process given by the stochastic differential equation where B denotes a standard Brownian motion. Since a Bessel process is non-ergodic, we transform it into a stationary and ergodic process by adding a mean reverting term with speed of mean reversion α > 0 in the drift, which we call modified Bessel process in the following. The two process are then related by the well-known spacetime transformation of a Bessel process. Since the eigenfunctions of the associated diffusion operator of the modified Bessel process are known, we base our martingale estimation function on these eigenfunctions and follow the lines of [ks1999].
For the estimating function based on the first eigenfunction we obtain an explicit formula for the estimator, which only depends quadratically on the observations. We see that the estimator coincides with the one of a linear martingale estimation function for the Cox Ingersoll Ross process, which is the square of the modified Bessel process. We discuss optimality in the sense of Godambe and Heyde. Note that in [or1997] also local asymptotic normality of the Cox Ingersoll Ross process for ϑ > 0 was established.
Furthermore, we consider martingale estimating functions based on the first two eigenfunctions and discuss the improvement of the asymptotic variance. In this case we do not get an explicit estimator anymore.
Note that our results for the Bessel process may also be used to estimate the multiplicity parameter k of a one-dimensional Dunkl process, a special jump diffusion given by the generator By the last term in the generator we see that the associated process possesses jumps due to a reflection, which lead to a sign change. Hence, the modulus of this Dunkl process is a Bessel process with dimensionality parameter k − 1/2, cf. [cgy2008]. For the Dunkl process the multiplicity parameter is of special interest, since it determines the jump activity, namely for k ≥ 0 a Dunkl process has a finite jump activity, whereas for k < 1/2 we have infinite jump activity. The paper is organised as follows: in Section 2 we collect the basic facts on the processes, Section 3 is devoted to martingale estimation functions based on the first eigenfunction and Section 4 to estimators based on two eigenfunctions.

Basic results on Bessel processes and a stationary modification
In this section we introduce the basic results on the underlying diffusions, which we will need in the following for the theory of martingale estimation functions. Our aim is to estimate the parameter ϑ ∈ Θ ⊂ (− 1 2 , ∞) of a classical one-dimensional Bessel process. Since a Bessel process is non-ergodic and most results on parameter estimation for diffusions are developed for ergodic diffusions, we start by introducing a modification of a Bessel process which is ergodic.
We consider the stochastic differential equation for a Brownian motion B, some fixed α > 0 and the parameter of interest ϑ ∈ Θ ⊂ (− 1 2 , ∞). The equation (2.1) is similar to the equation defining a Bessel process except for the drift term −αX t dt, which we add to ensure ergodicity and stationarity.
In order to determine the density of (X t ) t≥0 , we consider the space time transformation for a Bessel process (Y t ) t≥0 with index ϑ, which immediately follows by Itô's formula. Therefore, we derive the distribution of (X t ) t≥0 by using the well-known distribution of the Bessel process (Y t ) t≥0 , namely is the Bessel function with index ϑ (see for instance [imk1974]). Hence, we obtain We denote the density of X ∆ with starting point x by p ϑ (x, ·, ∆) and the distribution of X ∆ by P ϑ . In the following, we check that (X t ) t≥0 is indeed stationary and ergodic and determine the invariant measure. The density of the scale measure for a fixed ξ ∈ (0, ∞) is defined as Note that, due to the singularity in the drift, we initially have to consider some positive interior point ξ.
From this we may deduce that (X t ) t≥0 is ergodic as we see that the conditions As the invariant measure is defined via the scale measure m( dx) := 1 s(x) dx, we obtain by a straight forward calculation that the density of the invariant probability measure is given by on (0, ∞) with respect to the Lebesgue measure. For the calculation of the asymptotic variance we will need the symmetric distribution Q ϑ ∆ of two consecutive observations X (i−1)∆ and X i∆ on (0, ∞) 2 . It is given by

Martingale estimating functions based on eigenfunctions
In this section we proceed similarly to [bs1995] and [ks1999] to construct martingale estimation functions for our parameter of interest ϑ. The concepts in these papers are based on ergodic diffusions. As Bessel processes are non-ergodic we constructed the ergodic and stationary version in (2.1). Let X ∆ , . . . , X n∆ be discrete observations of the process. We consider the eigenfunctions of the generator which are the solutions of L ϑ φ η = −λ η φ η given by by Itô's formula. Consequently, we may use the general theory on estimators based on eigenfunctions given in [ks1999]. However, in our case we may calculate the involved quantities and obtain explicit results. For the first eigenfunction φ 1 (x, ϑ) = 1 − αx 2 ϑ+1 we consider the estimator based on the martingale estimating function The unique solution of G n ( ϑ n ) = 0 is Now, we may deduce consistency and asymptotic normality along the same lines as for general martingale estimating functions.

By using an explicit formula of the conditional mean, we conclude
Applying these formulas we establish Let us discuss the results. Looking at the asymptotic variance we see that it decreases when α∆ is increasing. This seems surprisingly at the first glance, since it implies that the asymptotic variance decreases when the distance between observations increases, as we keep the mean reverting parameter α fixed. On the other hand, keeping in mind that equidistant observations for the stationary version of the Bessel process means the distance between two observations of the underlying Bessel process is exponentially growing. This leads to a fast growing observation interval, capturing the non-stationary behaviour of the original Bessel process. Furthermore, we see that the asymptotic variance tends to infinity as the mean-reverting parameter tends to zero.
Having a closer look at the estimator, we see that it only depends on the square of the observations, hence we could reformulate our problem and consider the squared process Y t := X 2 t . Itô's formula yields dY t = 2 Y t dB t + (2ϑ + 2 − 2αY t ) dt, an equation describing a Cox Ingersoll Ross process. We consider now the canonical linear martingale estimating function For ϑ > − 1 2 the unique solution of G n ( ϑ n ) = 0 is again Hence, we see that the two estimators coincide. In 3.1 we have already established the consistence and asymptotic normality of ϑ n . The next step is to search for the optimal asymptotic variance by using estimators of the form where g i−1 is σ(X ∆ , . . . , X n∆ ) measurable and continuously differentiable. Considering this second approach via linear martingale estimating functions for the squared process, allows us easily to determine this optimal estimator, cf. [h1988], [gh1987]. By [bs1995, (2.10)] the optimal estimator is given by where φ is the conditional variance of X i∆ . Unfortunately, the equation is not explicitly solvable with respect to ϑ. However, we can nevertheless determine the improvement in the asymptotic variance. Following again the same lines as [bs1995, Theorem 3.2], we have to establish the finiteness of the reciprocal of the asymptotic variance. Consequently, we can deduce that a lower bound of the optimal variance is given by ϑ 0 + 1. Figure 1 shows the asymptotic behaviour of the 10.000 simulated optimal estimator (triangles) and ϑ n (dots) for n = 1.000. The solid line corresponds to the calculated asymptotic information of ϑ n in 3.1. The dotted line represents our computed bound above. As the lines nearly touch around ∆ = 3, the improvement of the optimal estimator quickly tends to zero. Starting from the value ∆ = 1 the simulated asymptotic information is almost the same for both estimators. Beforehand, the improvement is clearly visible but we do not want to maintain such a high variance as we can choose the value of α∆ such that the asymptotic variance is close to the lower bound.
We take a closer look at the asymptotic variance of ϑ n , which decreases monotonously in α∆: Due to the fast convergence to the lower bound ϑ 0 +1, we can for practical purposes restrict ourselves to the estimator ϑ n and hence have an explicit estimator.

Estimator based on two eigenfunctions
Now, we try to improve the asymptotic variance further by considering martingale estimating functions based on two eigenfunctions. Yet, this approach suffers from the drawback that we do not get an explicit estimator anymore.
We consider where β 1 and β 2 are continuously differentiable functions only depending on ϑ. Under suitable conditions on the interplay between the weights β i and the eigenfunctions, we can easily achieve a consistent and asymptotic normal estimator.