DEVIATION INEQUALITIES AND MODERATE DEVIATIONS FOR ESTIMATORS OF PARAMETERS IN AN ORNSTEIN-UHLENBECK PROCESS WITH LINEAR DRIFT

Some deviation inequalities and moderate deviation principles for the maximum likelihood estimators of parameters in an Ornstein-Uhlenbeck process with linear drift are established by the logarithmic Sobolev inequality and the exponential martingale method.


Introduction
We consider the following Ornstein-Uhlenbeck process It is known thatθ T andγ T are consistent estimators of θ and γ and have asymptotic normality (cf. [15]).
For γ ≡ 0 case, Florens-Landais and Pham( [9]) calculated the Laplace functional of ( T 0 X t d X t , T 0 X 2 t d t) by Girsanov's formula and obtained large deviations forθ T by Gärtner-Ellis theorem. Bercu and Rouault ( [1]) presented a sharp large deviation forθ T . Lezaud ([14]) obtained the deviation inequality of quadratic functional of the classical OU processes. We refer to [8] and [11] for the moderate deviations of some non-linear functionals of moving average processes and diffusion processes. In this paper we use the logarithmic Sobolev inequality (LSI) to study the deviation inequalities and the moderate deviations ofθ T andγ T for γ = 0 case.

Main results
Throughout this paper, let λ T , T ≥ 1 be a positive sequence satisfying Theorem 1.1. There exist finite positive constants C 0 , C 1 , C 2 and C 3 such that for all r > 0 and all In this theorem and the remainder of the paper, all the constants involved depend on θ , γ and the initial point x.
T ≥ 1 satisfies the large deviation principle with speed λ T and rate function I 1 (u) = u 2 4θ , that is, for any closed set F in , and open set G in , T ≥ 1 satisfies the large deviation principle with speed λ T and rate function I 2 (u) = θ u 2 2(θ +2γ 2 ) , that is, for any closed set F in , and open set G in , In γ = 0 case, the deviation inequalities of quadratic functionals of the classical OU process are obtained in [14]. For the large deviations and the moderate deviations ofθ T , we refer to [1], [9] and [11]. The proofs of Theorem 1.1 and Theorem 1.2 are based on the LSI with respect to L 2 -norm in the Wiener space and Herbst's argument (cf. [10], [12]).

Deviation inequalities
In this section, we give some deviation inequalities for the estimatorsθ T andγ T by the logarithmic Sobolev inequality and the exponential martingale method. For deviation bounds for additive functionals of Markov processes, we refer to [3] and [18].

Moments
It is known that the solution of equation (1.1) has the following expression: From this expression, it is easily seen that for any t ≥ 0, and there exist finite positive constants L 1 and L 2 such that for all 0 ≤ α ≤ θ 2 /4 and T ≥ 1, Proof. For any 0 ≤ α ≤ θ 2 /4, set κ = θ 2 − 2α. Then by Girsanov theorem, we have we complete the proof of the lemma.

Logarithmic Sobolev inequality
Since the LSI with respect to the Cameron-Martin metric does not produce the concentration inequality of correct order in large time T for the functionals in order to get the concentration inequality of correct order for the functionals F (X ), as pointed out by Djellout, Guillin .
Denote by C 1 b (W /L 2 ) the space of all bounded function f on W , differentiable with respect to the L 2 -norm, such that ∇ f is also continuous and bounded from W equipped with L 2 -norm to L 2 ([0, T ], ). Applying Theorem 2.3 in [10] to the Ornstein-Uhlenbeck process with linear drift, we have where the entropy of f 2 is given by
Then for any g ∈ L 2 ([0, T ], ), and so by (2.9), we have Letting n → ∞ and by Lemma 2.1, we get and so the conclusions of the lemma hold by Theorem 2.7 in [12] and Tμ 2 T ≤ T 0 X 2 t d t.

Deviation inequalities
it is easily to get from Chebyshev inequality, for any r > 0, where we used (2.7).

Lemma 2.3.
There exist finite positive constants C 0 , C 1 , C 2 such that for all r > 0 and all T ≥ 1, In particular, there exist finite positive constants C 0 , C 1 , C 2 such that for all r > 0 and all T ≥ 1, Proof. We only prove the first inequality. By Lemma 2.2 and Lemma 2.1, there exist finite positive constants L 1 and L 2 such that for all T ≥ 1, for any |α| ≤ θ 2 /4, Therefore, by Chebyshev inequality, for any r > 0, T ≥ 1 and |α| ≤ θ 2 /4, we obtain the first inequality of the lemma from the above estimates.

Lemma 2.4.
There exist finite positive constants C 0 , C 1 and C 2 such that for all r > 0 and all T ≥ 1, Proof. Since for any r > 0 and T ≥ 1, , by (2.12) and W T ∼ N (0, T ), we get

Proof of Theorem 1.1
We only show the first inequality. The second one is similar. Bŷ T for any r > 0 and T ≥ 1, Therefore, by Lemmas 2.3, 2.4 and 2.5, we obtain the first inequality of the theorem.

Moderate deviations
In this section, we show Theorem 1.2. By (1.2) and (1.3), we have the following estimates (2). For any δ > 0, Proof. (1). We only give the proof of the third assertion in (1). The rest is similar. For any L > 0, Hence, Letting L → ∞, we obtain the third conclusion.