Hypothesis Testing in a Fractional Ornstein-Uhlenbeck Model

Consider an Ornstein-Uhlenbeck process driven by a fractional Brownian motion. It is an interesting problem to find criteria for whether the process is stable or has a unit root, given a finite sample of observations. Recently, various asymptotic distributions for estimators of the drift parameter have been developed.We illustrate through computer simulations and through a Stein’s bound that these asymptotic distributions are inadequate approximations of the finite-sample distribution formoderate values of the drift and the sample size.We propose a newmodel to obtain asymptotic distributions near zero and compute the limiting distribution. We show applications to regression analysis and obtain hypothesis tests and their asymptotic power.


Introduction
Stability properties of the ordinary differential equation x t θx t depend on the sign of the parameter θ: the equation is asymptotically stable if θ < 0, neutrally stable if θ 0, and instable if θ > 0. These stability results carry over to the stochastic process X t X 0 θ t 0 X s ds Z t , 1.1 driven by noise Z Z t . When the value of θ is not known and a trajectory of X X t is observed over a finite time interval t ∈ 0, T , a natural problem is to develop the zero-root test, that is, a statistical procedure for testing the hypothesis θ 0 versus one of the possible alternatives θ / 0, θ > 0, or θ < 0. While the classical solution to this problem is wellknown to use the maximum likelihood estimator MLE of the parameter θ as the test statistic , further analysis is necessary because the exact distribution of the MLE is usually too complicated 2 International Journal of Stochastic Analysis to allow an explicit computation of either the critical region or the power of the test. More specifically, an approximate distribution of the MLE must be introduced and investigated, both in the finite-sample asymptotic and in the limit T → ∞. There are other potential complications, such as when MLE is not available e.g., if Z is a stable Lévy process, see 1 or when the MLE is difficult to implement numerically e.g., if Z is a fractional Brownian motion, see 2 .
The objective of this work is the analysis and implementation of the zero root test for 1.1 when Z W H , the fractional Brownian motion with the Hurst parameter H, and 1/2 ≤ H < 1. When 0 < H < 1/2, the integral transformation of Jost While the exact distribution of θ T,1/2 is not known, the following asymptotic relations hold as T → ∞: where N is the normal distribution, Ψ 1/2,0 is from 1.7 with H 1/2 and b 0, and C 1 is the standard Cauchy distribution with probability density function 1/ π 1 x 2 , x ∈ R. While 1.10 suggests that the distribution Ψ 1/2,0 can be used to construct an asymptotic zero-root test, it turns out that neither 1.9 nor 1.11 is a good choice for analyzing the power of the resulting test for small values of the product θT. There are two reasons: a both 1.9 and 1.11 suggest that the product θT should be sufficiently large for the corresponding approximation to work; b it follows from 1.9 -1.11 that the limit distribution of the appropriately normalized residual θ T,1/2 − θ has a discontinuity near θ 0, while, by 1.6 , the distribution of θ T,1/2 − θ is a continuous function of θ for each fixed T > 0. Further discussion of the finite sample statistical inference is in Sections 2 and 3. In particular, Table 1 and Figure 1 in Section 3 provide some numerical results when θ < 0.
It is therefore natural to derive a different family of asymptotic distributions, one that depends continuously on the parameter θ near θ 0. To this end, let θ θ T depend on the observation time T and lim T → ∞ θ T 0. Then, for each fixed T , equality 1.4 still defines the fractional Ornstein-Uhlenbeck process, but the asymptotic behavior of the estimator 1.8 changes.
The following is the main result of the paper. The proof is given in Section 3. The almost sure convergence can be shown using the law of iterated logarithm. To proof 1.12 , we first show asymptotic properties of individual terms of the estimator θ T,H . Finally, we combine these results using the continuous mapping theorem.

4
International Journal of Stochastic Analysis  An alternative least-squares type estimator has been considered in 5 and is given by The stochastic integral in 1. 13 where c ∈ R and 1 F 1 a, b, z is Kummer's hypergeometric function, see Section 3 for details.
International Journal of Stochastic Analysis 5 Theorem 1.2. Assume that 1/2 ≤ H < 1, and let θ θ T be a family of parameters such that lim T → ∞ Tθ T c for some c ∈ R. Then, as T → ∞, θ T,H → 0 with probability one and The proof is given in Section 3. The almost sure convergence can be shown using the law of iterated logarithm. To proof 1.15 , we first find a different representation of θ T,H . Then, we use asymptotic distributions of individual terms and combine these results using the continuous mapping theorem.

Strong Consistency and Large-Sample Asymptotic
In this section, we study the asymptotic behavior as T → ∞ of estimator θ T,H given by 1.6 . First, we show that it is a strongly consistent estimator of the parameter θ ∈ R. Consistency is a minimum requirement for any statistic to be of practical use for estimating parameter θ. Moreover, we derive its rate of convergence and corresponding limit theorems. Finally, we illustrate some estimation problems if θT is small through a Stein's bound and computer simulations.

Strong Consistency
We show that the estimator θ T,H is a strongly consistent estimator of θ, that is, for all θ ∈ R, with probability one. Proof. If θ < 0 and X H 0 0, then, by 6, Lemma 3.3 , with probability one; analysis of the proof shows that 2.2 also holds when with probability one, see 6 , the remark after the proof of Theorem 3.4 . Then, 2.1 follows from 2.2 and 2.3 .

International Journal of Stochastic Analysis
If θ > 0 and X H 0 0, then, by 7, Theorem 1 , with probability one; analysis of the proof shows that 2.4 also holds when X H 0 / 0. Also, by 7, Lemmas 2 and 3 , the finite limit exists with probability one. Therefore, all with probability one, and 2.2 follows. If θ 0, then X H t X H 0 W H t , and the law of iterated logarithm for selfsimilar Gaussian processes 8, Corollary 3.1 implies that with probability one for every ε > 0 and lim sup

Convergence Rates and Asymptotic Distributions
We have the following asymptotic distributions and convergence rates of θ T,H .
and η 1 and η 2 are independent standards normally distributed.  Case θ > 0: from 7, Theorem 5 , with the obvious modification to nonzero initial condition, e θT A T → 2θη 1 / η 2 X H 0 b H , with b H defined in 2.12 . Moreover, using the first convergence in 2.6 and 1/H > 1,
Case θ 0: the convergence in 2.10 follows from Theorem 1.1, which is proved later.

A Stein's Bound
While both 2.9 and 2.11 suggest that the rate of convergence is determined by the product |θ|T , more precise estimates are possible when θ < 0 and H 1/2. If H 1/2, then 1.8 implies that where W W 1/2 is the standard Brownian motion, X X 1/2 is the corresponding Ornstein-Uhlenbeck process, and θ T θ T,1/2 . In the following, we show that, when θ < 0 and H 1/2, the rate of convergence of a the numerator of 2.16 to the normal distribution and b the denominator of a constant indeed depends on how large the term |θ|T is. For a , we use elements of Stein's method on Wiener chaos see 9 . To simplify the notations, we switch from θ to −θ and assume zero initial condition, that is, we consider where θ > 0. For random variables X, Y on Ω, F, P , we define the total variation distance

2.18
International Journal of Stochastic Analysis 9 Note that T 2θ where I 2 f s, t 2 T 0 t 0 f s, t dW s dW t , denotes the iterated Wiener integral for symmetric square integrable functions f s, t . Then one can see that the numerator converges to a normal distribution, and the denominator converges to a constant, both almost surely and in mean square. By an estimate from 10, Section 2 , we get, using an application of Chebyshev's inequality, Thus the convergence of the denominator depends on the size of θT. For the numerator, we need some additional computations to get the rate of convergence to the normal. Note that F T is in the second Wiener chaos of W, see 5 , Section 1.1.2 for Wiener chaos for the white noise case. Hence, by 9 , Theorem 1.5 , we have where Z is a generic standard normally distributed random variable.
We have 2θ .

2.24
Combining 2.26 , 2.28 , and 2.23 , we finally conclude that An explicit value of C can be recovered from the above computations.

Computer Simulations
Both 2.22 and 2.29 suggest that the distribution of θ T,H will be rather different from normal for moderate values of 1/ √ θT. This conclusion is consistent with Monte Carlo simulations: if θ 1 and T 200 so that 1/ √ θT ≈ 1/14, moderate indeed , then the normality assumption for θ T,H is rejected at the significance level 5%, see Table 1.
In the next section, we study the asymptotic distribution of the statistic θ T,H and obtain a better finite-sample distribution approximation of θ T,H .

Finite-Sample Approximation and Hypothesis Testing
In this section, we develop approximations of the finite-sample distribution of the estimator which are different from 2.9 and 2.11 . The approximate distribution is continuous as a function of the suitable parameter and, according to Monte Carlo simulations, works well when 2.9 and 2.11 do not.
As a motivation, recall an analogous result for the first-order stochastic difference equation

3.2
It is known that α T is consistent estimator of α, that is, lim N → ∞ α N α in probability. Moreover, the asymptotic distribution as N → ∞ of α n is given by .3 has been proven in 11 , and 3.4 and 3.5 in 12 . Several authors deal with asymptotic distributions in the case that α is near 1, which has been extensively studied in 13-15 . The idea is to choose the parameter c according to where N is the sample size. Note that this family of parameters satisfy lim N → ∞ α N 1 and lim N → ∞ N α N − 1 c, where c < 0 corresponds to stationary case, c 0 corresponds to the unit root, and c > 0 corresponds to explosive case. The distribution of N α N converges to a functional of the Ornstein-Uhlenbeck process, see 13, Theorem 1 b : International Journal of Stochastic Analysis where Ψ 1/2,c is given in 1.7 with H 1/2, leading to a better asymptotic distribution than 3.3 and 3.5 for moderate values of α and N. The results in Section 2 suggest that, whenever the product |θ|T is small, continuoustime analogues of 3.6 and 3.7 are necessary, which needs a better understanding of stochastic integration with respect to the fractional Brownian motion. This understanding is necessary both to further analyze 1.6 and to establish the connection between 1.6 and 1.13 . Similar to 5, 16, 17 , we follow the Malliavin calculus approach.

Stochastic Integration with respect to Fractional Brownian Motion
As before, denote that W H W H t is a fractional Brownian motion with index H ∈ 1/2, 1 . It can be shown that W H has stationary increments, and it is self-similar, in the sense that for every c > 0, Assume furthermore that the sigma-field F is generated by W H t . Let E be the set of realvalued step functions on 0, T , and let H be the real separable Hilbert space defined as the closure of E with respect to the scalar product where f ∈ C ∞ p R n , the space of infinitely differentiable functions f : R n → R, where f and all its derivatives have at most polynomial growth.
Define the derivative operator of such F as the H-valued random variable The derivative operator D H is a closable unbounded operator from L p Ω, F, P to L p Ω, F, P; H for any p ≥ 1. Define as D H,1,p H , p ≥ 1, the closure of S with respect to the norm 3.13 For an element u ∈ dom δ H , we can define δ H u through the relationship Let f, g : 0, T → R be Hölder continuous functions of order α ∈ 0, 1 , and β ∈ 0, 1 respectively, with α β > 1. Young 19 proved that the Riemann-Stieltjes integral now known as the Young integral T 0 f s dg s exists. Accordingly, for H > 1/2, we can define the pathwise Young integral for any process u {u t , 0 ≤ t ≤ T } that has Hölder-continuous paths of order where 0 t 0 ≤ · · · ≤ t n T is any partition such that max i |t i 1 − t i | → 0 as n → ∞. In 17, Theorem 12 , it is shown that using the right endpoints such as u t i 1 in 3.19 does not change the limit. Then as in 6, 7 , we have the following relation between the two integrals and the Malliavin derivative:

Asymptotic Distribution of the Statistics
Let θ θ T depend on the observation time T and lim T → ∞ θ T 0. In analogy to the discrete time case 3.6 , we make the assumption that θ θ T depends on the observation time interval T , so that for some real number c. The parameter c plays the same role as in 3.6 . The particular form of θ T , for example, θ T c/T , will affect the finite-sample distribution of the least-squares estimator but, as long as 3.22 holds, it will not matter in the limit T → ∞. Strictly speaking, now we should be writing X H t, T , but, for the sake of simplicity of notations, we will omit the explicit dependence of X H on T . Lemma 3.1. As T → ∞, one has the following asymptotic distributions:  where It is shown in 6 that θ T is a strongly consistent estimator of θ < 0, that is, lim T → ∞ θ T θ with probability one. Reference 7 implied strong consistency for the case θ > 0 after a slight modification of the estimator.
In the case H 1/2, estimator 1.6 is also the maximum likelihood estimator of the parameter θ. Denote by P θ T the measure generated by the Ornstein-Uhlenbeck process Z {Z t , 0 ≤ t ≤ T }, Z t t 0 e θ t−s dW 1/2 s in the space of continuous functions C 0, T . Then, the measures P 0 T and P θ T are equivalent, and the likelihood function is given by

3.43
Maximizing the density with respect to θ leads to 1.6 . An extension of this result to secondorder differential equations is available in 22 . We cannot use θ T,H in practice for two reasons. First, there is no way to compute the divergence integral T 0 X H t dX H t given observations X H t , 0 ≤ t ≤ T . Second, the alternative representation we obtain in this section see Lemma 3.3 depends on the unknown parameter θ and, therefore, cannot be used to compute the value of θ T,H . Nonetheless, the finite-sample asymptotic for this estimator is an interesting subject to investigate, and we can also see similarities to θ T,H .
In the following, let 1 F 1 a, b, z be Kummer's confluent hypergeometric function see 23, Chapter 13 , which is given by 1 F 1 a, b, z ∞ n 0 a n z n b n n! , 3.44 where a n a a 1 a 2 · · · a n − 1 .