Estimating functions for noisy observations of ergodic diffusions

Abstract

In this article, general estimating functions for ergodic diffusions sampled at high frequency with noisy observations are presented. The theory is formulated in terms of approximate martingale estimating functions based on local means of the observations, and simple conditions are given for rate optimality. The estimation of the diffusion parameter is faster than the estimation of the drift parameter, and the rate of convergence is classical for the drift parameter but not classical for the diffusion parameter. The link with specific minimum contrast estimators is established, as an example.

Appendix

Define the following random variables which appear in some expansions:

$$\begin{aligned} \zeta _{j+1,N}= & {} \frac{1}{p_N} \sum _{i=0}^{p_N - 1} \int _{j \Delta _N + i \delta _N}^{(j+1) \Delta _N} d B_s, \end{aligned}$$

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$$\begin{aligned} \zeta _{j+2,N}'= & {} \frac{1}{p_N} \sum _{i=0}^{p_N - 1} \int _{(j+1) \Delta _N}^{(j+1) \Delta _N + i \delta _N} d B_s. \end{aligned}$$

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Notice that

$$\begin{aligned} \zeta _{j+1,N}= & {} \frac{1}{p_N} \sum _{k=0}^{p_N - 1} (k+1) \int _{j \Delta _N + k \delta _N}^{j \Delta _N + (k+1) \delta _N} dB_s, \end{aligned}$$

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$$\begin{aligned} \zeta _{j+2,N}'= & {} \frac{1}{p_N} \sum _{k=0}^{p_N - 1} (p_N - 1 - k) \int _{(j+1)\Delta _N + k \delta _N }^{(j+1) \Delta _N + (k+1) \delta _N} dB_s. \end{aligned}$$

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Lemma 4

The random variables $\zeta _{j+1,N}$ and $\zeta _{j+1,N}'$ are ${{\mathcal {G}}}_{(j+1) \Delta _N}$-mesurable, and $\zeta _{j+2,N}'$ is independent of ${{\mathcal {G}}}_{(j+1)\Delta _N}$. Moreover,

$$\begin{aligned} {{\mathbb {E}}}(\zeta _{j,N} | {{\mathcal {G}}}_j^N )= & {} 0 \\ {{\mathbb {E}}}(\zeta _{j+1,N}' | {{\mathcal {G}}}_j^N )= & {} 0 \end{aligned}$$

$$\begin{aligned} {{\mathbb {E}}}((\zeta _{j+1,N})^2 | {{\mathcal {G}}}_j^N )= & {} \Delta _N \left( \frac{1}{3} + \frac{1}{2 p_N} + \frac{1}{6 p_N^2} \right) \\ {{\mathbb {E}}}((\zeta _{j+1,N}')^2 | {{\mathcal {G}}}_j^N )= & {} \Delta _N \left( \frac{1}{3} - \frac{1}{2 p_N} + \frac{1}{6 p_N^2} \right) \\ {{\mathbb {E}}}(\zeta _{j+1,N} \zeta _{j+1,N}' | {{\mathcal {G}}}_j^N )= & {} \frac{\Delta _N}{6} \left( 1 -\frac{1}{p_N^2} \right) \end{aligned}$$

In Favetto (2014), several properties of $Y_{\bullet }^j$ and some results of convergence in probability for functionals of the blocks have been established. Recall that the random variables $\zeta _{j,N}$ and $\zeta _{j+1,N}'$ are defined in (4.8), and Lemma 6 is used to establish convergence in probability results.

Proposition 1

Under (A1), we have for $j \le k_N - 1$,

$$\begin{aligned} Y_{\bullet }^{j} - X_{j \Delta _N} = \sigma (X_{j \Delta _N}) \sqrt{\Delta _N} \xi _{j,N}' + e_{j,N}' + \rho \varepsilon _{\bullet }^j \end{aligned}$$

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with $|{{\mathbb {E}}}(e_{j,N}' |{{\mathcal {H}}}_j^N )| \le c \Delta _N (1 + |X_{j \Delta _N}|)$ and

$$\begin{aligned} {{\mathbb {E}}}( {e_{j,N}'}^2 | {{\mathcal {H}}}_j^N ) \le c \Delta _N^2 (1 + |X_{j \Delta _N}|^4), \qquad {{\mathbb {E}}}( {e_{j,N}'}^4 | {{\mathcal {H}}}_j^N ) \le c \Delta _N^3 (1 + |X_{j \Delta _N}|^4). \end{aligned}$$

If moreover (A5) holds, for $k \le 8$, there exists $c > 0$ such that, for $j \le k_N - 1$:

$$\begin{aligned} {{\mathbb {E}}}\left( \left. |Y_{\bullet }^j - X_{j \Delta _N} |^k \right| {{\mathcal {H}}}_j^N \right) \le C \left( \Delta _N^{k/2} (1 + | X_{j \Delta _N}|^{k}) + \rho ^k {{\mathbb {E}}}\left( | \varepsilon _{\bullet }^j |^{k} \right) \right) . \end{aligned}$$

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Proposition 2

Under Assumptions (A1)-(A2) and (A5), we have

$$\begin{aligned} Y_{\bullet }^{j+1} - Y_{\bullet }^j = \Delta _N b(X_{j \Delta _N}) + \sigma ( X_{j \Delta _N}) (\zeta _{j+1,N} + \zeta _{j+2,N}') + \tau _{j,N} + \rho (\varepsilon _{\bullet }^{j+1} - \varepsilon _{\bullet }^j) \end{aligned}$$

where $\tau _{j,N}$ is ${{\mathcal {H}}}_{j+2}^N$-mesurable, and there exists a constant c such that

$$\begin{aligned}&\displaystyle | {{\mathbb {E}}}(\tau _{j,N} | {{\mathcal {G}}}_j^N ) | \le c \Delta _N ( \Delta _N (1 + |X_{j \Delta _N}|^3) + \rho ^2 \sqrt{{{\mathbb {E}}}( (\varepsilon _{\bullet }^j)^4) }) ,\\&\displaystyle {{\mathbb {E}}}(\tau _{j,N}^2 | {{\mathcal {G}}}_j^N ) \le c \Delta _N ( 1 + |X_{j \Delta _N}|^2 + \rho ^2 {{\mathbb {E}}}( (\varepsilon _{\bullet }^j)^2)) (\Delta _N (1 + |X_{j \Delta _N} |^4) + \rho ^2 \sqrt{{{\mathbb {E}}}( (\varepsilon _{\bullet }^j)^4)}) , \\&\displaystyle {{\mathbb {E}}}( \tau _{j,N}^4 | {{\mathcal {G}}}_{j}^N ) \le c ( 1 + |X_{j \Delta _N}|^4 + \rho ^4 {{\mathbb {E}}}( ( \varepsilon _{\bullet }^j)^4) ) (\Delta _N^4 (1 + |X_{j \Delta _N}|^4 )+ \rho ^4 \sqrt{ {{\mathbb {E}}}( (\varepsilon _{\bullet }^j )^8)}) ,\\&\displaystyle | {{\mathbb {E}}}( \tau _{j,N} \zeta _{j+1,N} | {{\mathcal {G}}}_j^N ) | \le c \Delta _N ( 1 + |X_{j \Delta _N}|^2 + \rho ^2 {{\mathbb {E}}}( (\varepsilon _{\bullet }^j)^2) ) ( \Delta _N ( 1 + |X_{j \Delta _N}|^4) + \rho ^2 \sqrt{{{\mathbb {E}}}((\varepsilon _{\bullet }^j)^4)}) , \\&\displaystyle | {{\mathbb {E}}}( \tau _{j,N} \zeta _{j+2,N}' | {{\mathcal {G}}}_j^N ) | \le c \Delta _N ( 1 + |X_{j \Delta _N}|^2 + \rho ^2 {{\mathbb {E}}}( (\varepsilon _{\bullet }^j)^2) ) ( \Delta _N ( 1 + |X_{j \Delta _N}|^4) + \rho ^2 \sqrt{{{\mathbb {E}}}((\varepsilon _{\bullet }^j)^4)}. \end{aligned}$$

Lemma 5

Assume (A1)-(A3). Let $f \in \mathcal {C}^1 ({{\mathbb {R}}}\times O )$, where O is an open neighbourhood of $\varTheta$, satisfy

$$\begin{aligned} \sup _{\theta \in \varTheta } \{ |f(x,\theta ) | + | \partial _x f (x, \theta )| + |\partial _{\theta } f(x,\theta )| \} \le C ( 1 + |x|) \end{aligned}$$

then:

$$\begin{aligned} \frac{1}{k_N} \sum _{j=0}^{k_N-1} f(X_{j \Delta _N}, \theta ) \underset{k_N \rightarrow \infty }{\longrightarrow } \nu _0 (f(.,\theta )) \end{aligned}$$

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uniformly in $\theta$, in probability.

Theorem 6

Assume (A1)-(A5).

• 1. If $\delta _N = p_N^{-\alpha }$ with $\alpha \in (1,2)$ ($\Delta _N = p_N^{1-\alpha }$) and (B1) ($\rho = \rho >0$), then

  $$\begin{aligned} \bar{Q}_N (f(.,\theta )) = \frac{1}{k_N \Delta _N} \sum _{j=1}^{k_N - 2} f (Y_{\bullet }^{j-1} , \theta ) (Y_{\bullet }^{j+1} - Y_{\bullet }^j)^2 \overset{\mathbb {P}}{\longrightarrow } \frac{2}{3} \nu _0 (f(., \theta ) \sigma ^2 ), \end{aligned}$$

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• 2. If $\delta _N = p_N^{-2}$ ($\Delta _N = \frac{1}{p_N}$) and (B1) ($\rho = \rho >0$), then

  $$\begin{aligned} \bar{Q}_N (f(.,\theta )) \overset{\mathbb {P}}{\longrightarrow } \frac{2}{3} \nu _0 (f(., \theta ) \sigma ^2 ) + 2 \rho ^2 \nu _0 (f (. , \theta )), \end{aligned}$$

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• 3. If $\delta _N = p_N^{ - \alpha }$, $\alpha \in (1,2]$ with (B2) ($\rho \rightarrow 0$), then

  $$\begin{aligned} \bar{Q}_N (f(.,\theta )) \overset{\mathbb {P}}{\longrightarrow } \frac{2}{3} \nu _0 (f (. , \theta ) \sigma ^2 ), \end{aligned}$$

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where all the convergences in probability are uniform in $\theta \in \varTheta$, as $N \rightarrow \infty$, with $\delta _N \rightarrow 0$, $p_N \rightarrow \infty$, $k_N \rightarrow \infty$, $\Delta _N \rightarrow 0$ and $N \delta _N \rightarrow \infty$.

To study convergences in probability, an auxiliary result from Genon-Catalot and Jacod (1993) is reproduced here.

Lemma 6

Let $\chi _j^N$, U be random variables, with $\chi _j^N$ being ${{\mathcal {G}}}_j^N$-measurable. The following two conditions:

$$\begin{aligned}&\sum _{j=0}^{k_N -1} {{\mathbb {E}}}(\chi _j^N | \mathcal {G}_{j-1}^N) \overset{{{\mathbb {P}}}}{\rightarrow } U , \\&\sum _{j=0}^{k_N - 1} {{\mathbb {E}}}((\chi _j^N)^2 | \mathcal {G}_{j-1}^N) \overset{{{\mathbb {P}}}}{\rightarrow } 0 \end{aligned}$$

imply $\sum _{j=0}^{k_N - 1} \chi _j^N \overset{{{\mathbb {P}}}}{\rightarrow } U$.