Prediction-based estimation for diffusion models with high-frequency data

Abstract

This paper obtains asymptotic results for parametric inference using prediction-based estimating functions when the data are high-frequency observations of a diffusion process with an infinite time horizon. Specifically, the data are observations of a diffusion process at n equidistant time points $\Delta _n i$, and the asymptotic scenario is $\Delta _n \rightarrow 0$ and $n\Delta _n \rightarrow \infty$. For useful and tractable classes of prediction-based estimating functions, existence of a consistent estimator is proved under standard weak regularity conditions on the diffusion process and the estimating function. Asymptotic normality of the estimator is established under the additional rate condition $n\Delta _n^3 \rightarrow 0$. The prediction-based estimating functions are approximate martingale estimating functions to a smaller order than what has previously been studied, and new non-standard asymptotic theory is needed. A Monte Carlo method for calculating the asymptotic variance of the estimators is proposed.

Appendices

Appendix A: Proofs

Proof of Lemma 3.2

The diffusion process $(X_t)$ is reversible under Condition 2.2, so by Theorems 2.4 and 2.6 in Genon-Catalot et al. (2000) $\left\Vert P_t^\theta f \right\Vert _2 \le \rho _X(t) \left\Vert f \right\Vert _2 = e^{-\lambda t} \left\Vert f \right\Vert _2$, for any $f \in \mathscr {L}^2_0(\mu _\theta )$ , where $\lambda >0$ denotes the spectral gap of $\mathcal {A}_\theta$. $\square$

Proof of Proposition 3.3

Let $U_\theta ^{(n)}(f)=\int _0^n P_t^\theta f \mathrm {d}t$. By Property P4 in Hansen and Scheinkman (1995), $U_\theta ^{(n)}(f) \in \mathcal {D}_{\mathcal {A}_\theta }$ for all $n \in \mathbb {N}$ and:

$$\begin{aligned} \lim _{n \rightarrow \infty } \mathcal {A}_\theta \left( U_\theta ^{(n)}(f)\right) = \lim _{n \rightarrow \infty } \left[ P_n^\theta f-f\right] = - f, \end{aligned}$$

where limits are w.r.t. $\left\Vert \cdot \right\Vert _2$. The latter equality holds, because $\left\Vert P_n^\theta f \right\Vert _2 \le \left\Vert f \right\Vert _2 e^{-\lambda n} \rightarrow 0$.

By Jensen’s inequality, Fubini’s theorem, and Lemma 3.2, $U_\theta ^{(n)}(f)$ converges to $U_\theta (f)$ in $\mathscr {L}^2(\mu _\theta )$ as $n \rightarrow \infty$:

$$\begin{aligned} \left\Vert U_\theta (f)-U_\theta ^{(n)}(f) \right\Vert _2^2= & {} \int _S \left( \int _0^\infty 1\{t \ge n\} \lambda ^{-1} e^{\lambda t} P_t^\theta f(x) \lambda e^{-\lambda t} \mathrm {d}t\right) ^2 \mu _\theta (dx) \\\le & {} \int _S \left( \int _0^\infty 1\{t \ge n\} \lambda ^{-2} e^{2\lambda t} \left( P_t^\theta f(x)\right) ^2 \lambda e^{-\lambda t} \mathrm {d}t\right) \mu _\theta (dx) \\= & {} \lambda ^{-1} \int _n^\infty e^{\lambda t} \cdot || P_t^\theta f ||^2_2 \mathrm {d}t \\\le & {} \lambda ^{-1} \left\Vert f \right\Vert ^2_2 \int _n^\infty e^{-\lambda t} \mathrm {d}t = \lambda ^{-2} \left\Vert f \right\Vert ^2_2 e^{-\lambda n} \rightarrow 0. \end{aligned}$$

Taking $n=0$, we obtain (15). Using that $\mathcal {A}_\theta$ is closed and linear, we conclude that $\mathcal {A}_\theta \left( U_\theta (f)\right) = \mathcal {L}_\theta \left( U_\theta (f)\right) = -f$; see, e.g., Property P7, Hansen and Scheinkman (1995). $\square$

Proof of Proposition 3.4

The proof is an application of the central limit theorem for martingales. For completeness and because we need to extend the result in a non-standard way later, we give the proof. First, note that:

$$\begin{aligned} \frac{1}{\sqrt{n\Delta _n}} \int _0^{n\Delta _n} f(X_s) \mathrm {d}s= & {} \frac{1}{\sqrt{n\Delta _n}} \sum _{i=1}^n \int _{(i-1)\Delta _n}^{i\Delta _n} f(X_s) \mathrm {d}s \\= & {} \frac{1}{\sqrt{n\Delta _n}} \sum _{i=1}^n \int _{(i-1)\Delta _n}^{i\Delta _n} \left[ f(X_s) - f(X_{t^n_{i-1}})\right] \mathrm {d}s + \sqrt{n\Delta _n} V_n(f), \end{aligned}$$

where we will show that:

$$\begin{aligned} \frac{1}{\sqrt{n\Delta _n}} \sum _{i=1}^n \int _{(i-1)\Delta _n}^{i\Delta _n} \left[ f(X_s) - f(X_{t^n_{i-1}})\right] \mathrm {d}s = o_{\mathbb {P}_0}(1). \end{aligned}$$

(36)

With $A_i:=\int _{(i-1)\Delta _n}^{i\Delta _n} \left[ f(X_s) - f(X_{t^n_{i-1}})\right] \mathrm {d}s$, Fubini’s theorem combined with Lemma B.2 implies that:

for a generic function $F(x;\theta _0)$ of polynomial growth in x. Since $n\Delta _n^3 \rightarrow 0$, it follows by Lemma 3.1 that:

Moreover, for all $k \ge 1$, Jensen’s inequality implies that:

$$\begin{aligned} |A_i|^k= & {} \Delta _n^k \left|\frac{1}{\Delta _n} \int _{(i-1)\Delta _n}^{i\Delta _n} \left[ f(X_s) - f(X_{t^n_{i-1}})\right] \mathrm {d}s \right|^k \\\le & {} \Delta _n^{k-1} \int _{(i-1)\Delta _n}^{i\Delta _n} |f(X_s) - f(X_{t^n_{i-1}})|^k \mathrm {d}s \le \Delta _n^k \sup _{u \in [0,\Delta _n]} |f(X_{t^n_{i-1}+u}) - f(X_{t^n_{i-1}})|^k, \end{aligned}$$

and, hence, by Lemma B.1:

The conclusion (36) now follows from Lemma 9 in Genon-Catalot and Jacod (1993).

To apply the central limit theorem for martingales, note that Proposition 3.3 and Itô’s formula applied to $U_0(f)$ imply that:

$$\begin{aligned} U _0(f)(X_t)= & {} U _0(f)(X_0) + \int _0^t \mathcal {L}_0(U_0(f))(X_s) \mathrm {d}s + \int _0^t \partial _x U_0(f)(X_s)b(X_s;\theta _0) dB_s \\= & {} U _0(f)(X_0) - \int _0^t f(X_s) \mathrm {d}s + \int _0^t \partial _x U_0(f)(X_s)b(X_s;\theta _0) dB_s, \end{aligned}$$

so

$$\begin{aligned} \frac{1}{\sqrt{n\Delta _n}} \int _0^{n\Delta _n} f(X_s) \mathrm {d}s = \frac{1}{\sqrt{n\Delta _n}} \int _0^{n\Delta _n} \partial _x U_0(f)(X_s)b(X_s;\theta _0) dB_s + o_{\mathbb {P}_0}(1). \end{aligned}$$

(37)

The stochastic integral is a true martingale under $\mathbb {P}_0$ and by the ergodic theorem:

$$\begin{aligned} \frac{1}{n\Delta _n} \int _0^{n\Delta _n} \left[ \partial _x U_0(f)(X_s)b(X_s;\theta _0)\right] ^2 \mathrm {d}s \xrightarrow {\mathbb {P}_0}\mu _0\left( [\partial _x U_0(f) b( \cdot ;\theta _0)]^2\right) . \end{aligned}$$

In conclusion:

$$\begin{aligned} \sqrt{n\Delta _n} V_n(f)= & {} \frac{1}{\sqrt{n\Delta _n}} \int _0^{n\Delta _n} \partial _x U_0(f)(X_s)b(X_s;\theta _0) dB_s + o_{\mathbb {P}_0}(1) \nonumber \\&\xrightarrow {\mathscr {D}_0}\mathcal {N}\left( 0,\mu _0\left( [\partial _x U_0(f) b( \cdot ;\theta _0)]^2\right) \right) , \end{aligned}$$

(38)

where convergence in law under $\mathbb {P}_0$ follows from the continuous-time martingale central limit theorem (e.g., Theorem 6.31 in Häusler and Luschgy (2015)) or the central limit theorem for martingale arrays (e.g., Theorem 3.2 in Hall and Heyde (1980)). The conditional Lyapunov condition can be verified as in the proof of Theorem 4.5.

The alternative expression for the asymptotic variance $\mathcal {V}_0(f)$ in (16) follows, because with $g = U_0(f)$ and $b_0(x) = b(x;\theta _0)$, it follows from Proposition 3.3 that:

$$\begin{aligned} 2 \mu _0(fg) = - \mu _0\left( \mathcal {L}_0(g^2)\right) + \mu _0\left( b_0^2 \left[ \frac{1}{2} (g^2)'' - g g'' \right] \right) = \mu _0 \left( (b_0g')^2 \right) , \end{aligned}$$

where we have used that $\mu _0(\mathcal {L}_0(g^2)) = 0$, see, e.g., Hansen and Scheinkman (1995), p. 774. $\square$

Proof of Theorem 4.2

Under the conditions of theorem, the function $\kappa$ is 1-1, and $\kappa ^{-1}$ is continuous. By Lemma 3.1, $V_n(f) \xrightarrow {\mathbb {P}_0}\kappa (\theta _0)$ as $n \rightarrow \infty$. We have assumed that $\theta _0 \in \mathrm{int} \, \Theta$, so $\kappa (\theta _0) \in \mathrm{int} \, \kappa (\Theta )$, and hence, $\mathbb {P}_0(V_n(f) \in \kappa (\Theta )) \rightarrow 1$ as $n \rightarrow \infty$.

When $V_n(f) \in \kappa (\Theta )$, ${\hat{\theta }}_n = \kappa ^{-1} (V_n(f))$ is the unique $G_n$-estimator. When $V_n(f) \notin \kappa (\Theta )$, we set ${\hat{\theta }}_n := \theta ^*$ for some $\theta ^* \in \Theta$. Then, ${\hat{\theta }}_n \xrightarrow {\mathbb {P}_0}\theta _0$ as $n \rightarrow \infty$, and by a Taylor expansion:

$$\begin{aligned} \sqrt{n\Delta _n} \left( {\hat{\theta }}_n - \theta _0\right) = \partial _\theta \kappa (\theta _0) \sqrt{n\Delta _n}V_n(f^*) + o_{\mathbb {P}_0}(1), \end{aligned}$$

so (19) follows from Proposition 3.4. $\square$

Proof of Lemma 4.4

To simplify the presentation, we define:

$$\begin{aligned} H_n(\theta ) = \frac{1}{n\Delta _n} \sum _{i=1}^n g(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta ), \end{aligned}$$

(39)

where $g=(g_1,g_2)^T$ is given by:

$$\begin{aligned} g_1(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta )= & {} f(X_{t^n_i})-\breve{a}_n(\theta )_0-\breve{a}_n(\theta )_1f(X_{t^n_{i-1}}) \end{aligned}$$

(40)

$$\begin{aligned} g_2(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta )= & {} f(X_{t^n_{i-1}}) g_1(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta ). \end{aligned}$$

(41)

As a first step, we verify the expansion (21) of $\breve{a}_n(\theta )$ in powers of $\Delta _n$. By Lemma B.2:

which implies that:

where $\left|R(\Delta _n;\theta ) \right| \le C(\theta )$ for a constant $C(\theta )>0$. This yields the $\Delta _n$-expansion:

$$\begin{aligned} \breve{a}_n(\theta )_1 = \frac{\mathbb {E}_\theta \left[ f(X_0)f(X_{\Delta _n})\right] -\left[ \mu _\theta (f)\right] ^2}{\mathbb {V}\mathrm{ar}_\theta f(X_0)} = 1 + \Delta _n K_f(\theta ) + \Delta _n^2 R(\Delta _n;\theta ), \end{aligned}$$

(42)

and, as a consequence:

$$\begin{aligned} \breve{a}_n(\theta )_0 = -\Delta _n K_f(\theta )\mu _\theta (f) + \Delta _n^2 R(\Delta _n;\theta ). \end{aligned}$$

(43)

This expansion of $\breve{a}_n(\theta )$ together with

implies that:

(44)

Hence, by Lemma 3.1:

where the contribution from the first term vanishes, because $\mu _0(\mathcal {L}_0f)=0$; see, e.g., Hansen and Scheinkman (1995).

To apply Lemma 9 in Genon-Catalot and Jacod (1993), it remains to show that:

(45)

From the expansions (42) and (43), it follows that:

$$\begin{aligned} \breve{\pi }_{i-1}(\theta ) = \breve{a}_n(\theta )_0 + \breve{a}_n(\theta )_1f(X_{t^n_{i-1}}) = f(X_{t^n_{i-1}}) + \Delta _n R(\Delta _n,X_{t^n_{i-1}};\theta ), \end{aligned}$$

which, in turn, yields the decomposition:

$$\begin{aligned}&{g^2_1(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta ) }\\&\quad = \left[ f(X_{t^n_i})-f(X_{t^n_{i-1}})\right] ^2 + 2 \left[ f(X_{t^n_i})-f(X_{t^n_{i-1}})\right] \Delta _n R(\Delta _n,X_{t^n_{i-1}};\theta ) \\&\qquad + \Delta _n^2 R(\Delta _n,X_{t^n_{i-1}};\theta ). \end{aligned}$$

Lemma B.1 implies that:

where we use that $n\Delta _n \rightarrow \infty$. Similarly:

and finally:

$$\begin{aligned} \frac{1}{n^2}\sum _{i=1}^n R(\Delta _n,X_{t^n_{i-1}};\theta ) \xrightarrow {\mathbb {P}_0}0, \end{aligned}$$

which together implies (45). Thus, by Lemma 9 in Genon-Catalot and Jacod (1993):

$$\begin{aligned} \frac{1}{n\Delta _n} \sum _{i=1}^n g_1(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta ) \xrightarrow {\mathbb {P}_0}K_f(\theta ) (\mu _\theta -\mu _0)(f). \end{aligned}$$

Similarly, for $g_2(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta )$, it follows easily from (44) that:

and hence:

Moreover, since $g^2_2(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta ) = f^2(X_{t^n_{i-1}})g^2_1(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta )$, we easily see that:

so the first conclusion of the lemma follows from Lemma 9 in Genon-Catalot and Jacod (1993).

To establish the limit of $\partial _{\theta ^T} H_n(\theta )$, we write:

$$\begin{aligned} H_n(\theta ) = \frac{1}{n\Delta _n}\sum _{i=1}^n Z_{i-1}\left[ f(X_{t^n_i})-Z_{i-1}^T \breve{a}_n(\theta )\right] , \end{aligned}$$

which implies that:

$$\begin{aligned} \partial _{\theta ^T} H_n(\theta ) = - \frac{1}{n\Delta _n}\sum _{i=1}^n Z_{i-1}Z_{i-1}^T \partial _{\theta ^T} \breve{a}_n(\theta ) = Z_n(f)A_n(\theta ), \end{aligned}$$

where $Z_n(f) := \frac{1}{n} \sum _{i=1}^n Z_{i-1}Z_{i-1}^T$ and $A_n(\theta ) := -\Delta _n^{-1} \partial _{\theta ^T} \breve{a}_n(\theta )$. By Lemma 3.1:

$$\begin{aligned} Z_n(f) \xrightarrow {\mathbb {P}_0}Z(f) = \left( \begin{array}{cc} 1 &{} \mu _0(f) \\ \mu _0(f) &{} \mu _0(f^2) \end{array}\right) , \end{aligned}$$

and applying the expansion (21):

$$\begin{aligned} A_n(\theta ) = \partial _{\theta ^T} \left( \begin{array}{c} K_f(\theta ) \mu _\theta (f) \\ -K_f(\theta ) \end{array}\right) + \Delta _n \partial _{\theta ^T}R(\Delta _n;\theta ) \rightarrow \partial _{\theta ^T} \left( \begin{array}{c} K_f(\theta ) \mu _\theta (f) \\ -K_f(\theta ) \end{array}\right) =: A(\theta ), \end{aligned}$$

which holds under the regularity assumption (23). Collecting our observations:

$$\begin{aligned}&\partial _{\theta ^T} H_n(\theta ) \xrightarrow {\mathbb {P}_0}Z(f)A(\theta )\\&\quad = \left( \begin{array}{cc} 1 &{} \mu _0(f) \\ \mu _0(f) &{} \mu _0(f^2) \end{array}\right) \left( \begin{array}{cc} \partial _{\theta _1}\left[ K_f(\theta )\mu _\theta (f)\right] &{} \partial _{\theta _2}\left[ K_f(\theta )\mu _\theta (f)\right] \\ -\partial _{\theta _1} K_f(\theta ) &{} -\partial _{\theta _2} K_f(\theta ) \end{array}\right) . \end{aligned}$$

To argue that the convergence is uniform over a compact subset $\mathcal {M}\subseteq \Theta$, note that:

$$\begin{aligned} \left\Vert \partial _{\theta ^T} H_n(\theta ) - Z(f)A(\theta ) \right\Vert \le \left\Vert Z_n(f)[A_n(\theta )-A(\theta )] \right\Vert + \left\Vert [Z_n(f)-Z(f)]A(\theta ) \right\Vert , \end{aligned}$$

and in particular:

$$\begin{aligned}&\sup _{\theta \in \mathcal {M}}\left\Vert \partial _{\theta ^T} H_n(\theta ) - Z(f)A(\theta ) \right\Vert \\&\quad \le \left\Vert Z_n(f) \right\Vert \sup _{\theta \in \mathcal {M}}\left\Vert A_n(\theta )-A(\theta ) \right\Vert + \left\Vert Z_n(f)-Z(f) \right\Vert \sup _{\theta \in \mathcal {M}}\left\Vert A(\theta ) \right\Vert . \end{aligned}$$

By continuity of norms, $\left\Vert Z_n(f) \right\Vert \xrightarrow {\mathbb {P}_0}\left\Vert Z(f) \right\Vert$ and $\left\Vert Z_n(f)-Z(f) \right\Vert = o_{\mathbb {P}_0}(1)$, so (25) follows by observing that:

$$\begin{aligned} \sup _{\theta \in \mathcal {M}}\left\Vert A_n(\theta )-A(\theta ) \right\Vert = \Delta _n \sup _{\theta \in \mathcal {M}} \left\Vert \partial _{\theta ^T} R(\Delta _n;\theta ) \right\Vert \le C(\mathcal {M}) \Delta _n \rightarrow 0, \end{aligned}$$

and using the continuity of $\theta \mapsto A(\theta )$. $\square$

Proof of Theorem 4.5

We continue with the notation (39)–(41) introduced above. Existence of a consistent sequence of $G_n$-estimators $(\hat{\theta }_n)$ follows from Theorem 2.5 in Jacod and Sørensen (2018), because the conclusions of Lemma 4.4 and the assumption that $W(\theta _0)$ is non-singular imply Condition 2.2 in Jacod and Sørensen (2018). The uniqueness result follows from Theorem 2.7 in Jacod and Sørensen (2018) under the identifiability condition $\gamma (\theta _0;\theta ) \ne 0$ for $\theta \ne \theta _0$. The function $\theta \mapsto \gamma (\theta _0;\theta )$ is called $G(\theta )$ in Jacod and Sørensen (2018) and is necessarily continuous.

Asymptotic normality when $n\Delta _n^3 \rightarrow 0$ follows from Theorem 2.11 in Jacod and Sørensen (2018). We only need to check that:

$$\begin{aligned} \sqrt{n\Delta _n} H_n(\theta _0) \xrightarrow {\mathscr {D}_0}\mathcal {N}_2(0,\mathcal {V}_0(f)). \end{aligned}$$

(46)

We apply the Cramér–Wold device to prove this weak convergence result, i.e., we must prove that for all $c_1,c_2 \in \mathbb {R}$:

$$\begin{aligned} C_n= & {} c_1 \cdot \frac{1}{\sqrt{n\Delta _n}} \sum _{i=1}^n g_1(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta _0) + c_2 \cdot \frac{1}{\sqrt{n\Delta _n}} \sum _{i=1}^n g_2(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta _0) \nonumber \\&\quad \xrightarrow {\mathscr {D}_0}\mathcal {N}\left( 0,\mu _0\left( \left[ \partial _x U _0(c_1f^*_1+c_2f^*_2) + c_2 f f' \right] ^2 b^2( \cdot ;\theta _0)\right) \right) . \end{aligned}$$

(47)

Reusing the expansions (42) and (43), we find that:

$$\begin{aligned} g_1(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta _0) = f(X_{t^n_i}) - f(X_{t^n_{i-1}}) + \Delta _n f^*_1(X_{t^n_{i-1}}) + \Delta _n^2 R(\Delta _n,X_{t^n_{i-1}};\theta _0), \end{aligned}$$

where $f^*_1$ is defined in Condition 4.3. Hence:

$$\begin{aligned}&{\frac{1}{\sqrt{n\Delta _n}} \sum _{i=1}^n g_1(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta _0)} \\&\quad = \frac{1}{\sqrt{n\Delta _n}} \sum _{i=1}^n \left[ f(X_{t^n_i}) - f(X_{t^n_{i-1}})\right] \\&\qquad + \sqrt{n\Delta _n} \cdot V_n(f^*_1) + (n\Delta _n^3)^{1/2} \cdot \frac{1}{n} \sum _{i=1}^n R(\Delta _n,X_{t^n_{i-1}};\theta _0) \\&\quad = \sqrt{n\Delta _n} \cdot V_n(f^*_1) + o_{\mathbb {P}_0}(1), \end{aligned}$$

because the first term in the expansion is a telesco** sum. Note that asymptotic normality for the first coordinate of the estimating function follows from Proposition 3.4. However, to obtain joint weak convergence, we need to consider the second coordinate too, which requires more work.

By Itô’s formula:

$$\begin{aligned} f(X_{t^n_i}) - f(X_{t^n_{i-1}}) = \Delta _n \mathcal {L}_0f(X_{t^n_{i-1}}) + A_i(\theta _0) + M_i(\theta _0), \end{aligned}$$

where

$$\begin{aligned} A_i(\theta )= & {} \int _{(i-1)\Delta _n}^{i\Delta _n} \left[ \mathcal {L}_\theta f(X_s) - \mathcal {L}_\theta f(X_{t^n_{i-1}})\right] \mathrm {d}s, \\ M_i(\theta )= & {} \int _{(i-1)\Delta _n}^{i\Delta _n} f'(X_s)b(X_s;\theta ) dB_s, \end{aligned}$$

and, hence, by applying the expansions (42) and (43) as above:

$$\begin{aligned}&{g_2(\Delta _n,X_{t^n_i},X_{t^n_{i-1}};\theta _0)} \\&\quad = f(X_{t^n_{i-1}}) A_i(\theta _0) + \Delta _n f^*_2(X_{t^n_{i-1}}) + f(X_{t^n_{i-1}}) M_i(\theta _0) + \Delta _n^2 R(\Delta _n,X_{t^n_{i-1}};\theta _0). \end{aligned}$$

A straightforward extension of the proof of (36) implies that:

$$\begin{aligned} \frac{1}{\sqrt{n\Delta _n}} \sum _{i=1}^n f(X_{t^n_{i-1}}) A_i(\theta _0) = o_{\mathbb {P}_0}(1), \end{aligned}$$

since $n\Delta _n^3 \rightarrow 0$ and as a consequence:

$$\begin{aligned} C_n = \sqrt{n\Delta _n} V_n(f^*) + \frac{1}{\sqrt{n\Delta _n}} \sum _{i=1}^n f(X_{t^n_{i-1}}) M^n_i(\theta _0) + o_{\mathbb {P}_0}(1), \end{aligned}$$

where $f^* = c_1 f_1^* + c_2 f_2^*$.

To gather the non-negligible terms, we argue as in (38) that:

which, in turn, yields the stochastic integral representation:

At this point, we can apply the central limit theorem for martingale difference arrays; see, e.g., Hall and Heyde (1980) or Häusler and Luschgy (2015). To shorten the notation in the following, we define:

and

$$\begin{aligned} h(x) = \left[ \partial _x U _0(f^*)(x) + f(x) f'(x)\right] ^2 b^2(x;\theta _0). \end{aligned}$$

First, by the conditional Itô’ isometry, Tonelli’s theorem, and Lemma B.2:

Moreover, for any $g \in \mathcal {C}^2_p(S)$ and $k \ge 2$, the Burkholder–Davis–Gundy inequality, Jensen’s inequality, Tonelli’s theorem, and Lemma B.2, respectively, imply that:

so based on the inequality

$$\begin{aligned} \left|Z_i \right|^3 \le _C \left|\int _{(i-1)\Delta _n}^{i\Delta _n} \partial _x U _0(f^*)(X_s)b(X_s;\theta _0) dB_s \right|^3 + |f(X_{t^n_{i-1}})|^3\left|\int _{(i-1)\Delta _n}^{i\Delta _n} f'(X_s)b(X_s;\theta _0) dB_s \right|^3, \end{aligned}$$

we conclude that:

Now, the martingale central limit theorem for triangular arrays implies (47), so (46) follows by the Cramér–Wold device. The alternative expression for the matrix $\mathcal {V}_0(f)$ follows, because by Proposition 3.4$\mu _0\left( [\partial _x U_0(g) b( \cdot ;\theta _0)]^2\right) = 2\mu _0\left( g U _0 (g)\right)$ for $g \in \mathscr {H}^2_0$, and because with $g_i = U_0(f_i^*)$ and $b_0(x) = b(x;\theta _0)$ it follows from Proposition 3.3 that:

$$\begin{aligned} \mu _0(f_1^*g_2 + f_2^*g_1) = - \mu _0\left( \mathcal {L}_0(g_1g_2)\right) + \mu _0\left( b_0^2 g_1' g_2' \right) = \mu _0 \left( b_0^2g_1'g_2' \right) , \end{aligned}$$

where we have used that $\mu _0(\mathcal {L}_0(g_1g_2)) = 0$, see, e.g., Hansen and Scheinkman (1995), p. 774. $\square$

Proof of Proposition 5.1

By the Cauchy–Schwarz inequality and the inequality (15):

$$\begin{aligned} \left|\mu _0\left( f^* U _0(f^*)\right) \right| \le \left\Vert f^* \right\Vert _2 \left\Vert U _0(f^*) \right\Vert _2 \le \frac{\left\Vert f^* \right\Vert _2^2}{\lambda _0}, \end{aligned}$$

where $\lambda _0>0$ denotes the spectral gap of $(X_t)$ under $\mathbb {P}_0$. Hence:

$$\begin{aligned} \text {AVAR}(\hat{\theta }_n)= \frac{2\, \mu _0\left( f^* U _0(f^*)\right) }{[\partial _{\theta }\mu _{\theta _0}(f)]^2} \le \frac{2 \, \mathbb {V}\mathrm{ar}_0f(X_0)}{\lambda _0 \, [\partial _{\theta }\mu _{\theta _0}(f)]^2}. \end{aligned}$$

$\square$

Appendix B: Moment expansions

The proofs in Appendix A rely on conditional moment expansions for diffusion models and the following results are essentially taken from Gloter (2000) and Florens-Zmirou (1989), respectively. In the sequel, $\theta \in \Theta$ is arbitrary and we assume for convenience that $0<\Delta <1$.

Lemma B.1

Let $f \in \mathcal {C}^1_p(S)$. For any $k \ge 1$, there exists a constant $C_{k,\theta }>0$, such that:

For completeness, we give a rough proof of the following lemma.

Lemma B.2

Suppose that $a(x;\theta ) \in \mathcal {C}_p^{2k,0}(S \times \Theta )$, $b(x;\theta ) \in \mathcal {C}_p^{2k,0}(S \times \Theta )$ and $f \in \mathcal {C}^{2(k+1)}_p(S)$ for some $k \ge 0$. Then:

Proof

We only consider $k=0$, the general case may be shown by induction; see Lemma 1.10, Sørensen (2012). By Itô’s formula:

$$\begin{aligned} f(X_{t+\Delta }) = f(X_t) + \int _t^{t+\Delta } \mathcal {L}_\theta f(X_s) \mathrm {d}s + \int _t^{t+\Delta } \partial _x f(X_s)b(X_s;\theta )dB_s, \end{aligned}$$

and since $\partial _x f$ and $b( \cdot ;\theta )$ are of polynomial, respectively, linear growth in x, the stochastic integral is a true $(\mathcal {F}_t)$-martingale w.r.t. $\mathbb {P}_\theta$ and:

Moreover, since $\mathcal {L}_\theta f$ is of polynomial growth in x:

$$\begin{aligned} |\mathcal {L}_\theta f(X_{t+u})| \le _C 1 + |X_t|^C + |X_{t+u}-X_t|^C, \end{aligned}$$

and hence:

by a simple application of Lemma B.1. $\square$