Local Asymptotic Normality Property for Ornstein–Uhlenbeck Processes with Jumps Under Discrete Sampling

Abstract

We address the issue of the local asymptotic normality property and the Fisher information for three characterizing parameters of Ornstein–Uhlenbeck processes with jumps under low frequency and high frequency discrete sampling with expanding observation window. The martingale method with the Kolmogorov backward equation and the Malliavin calculus are employed to derive explicit formulas for derivatives of the likelihood ratio function in the form of conditional expectation, which serve as essential tools for justifying the passage to the limit by the dominated convergence theorem. This approach makes it possible to carry out the proof without specifying the law of the jump component and without knowing the tail behaviors of the transition probability density and, as a consequence, to keep various types of jump structure within the scope of this article. The Fisher information under high-frequency sampling is essentially identical to the one for purely Gaussian Ornstein–Uhlenbeck processes due to the dominance of the Gaussian component over the jump component in the short time framework.

Appendix: Basic Facts on the Malliavin Calculus

We here summarize basic facts on the Malliavin calculus on the Gaussian space in brief. (For details and more notations, we refer the reader to the monograph [27] and [17].) Fix $T>0$ and we denote, as before, by $(\fancyscript{F}^1_t)_{t\in [0,T]}$ and $(\fancyscript{F}^2_t)_{t\in [0,T]}$ the filtrations generated respectively by $\{W_t:\,t\in [0,T]\}$ and $\{\mu ({\text{ d}}z,{\text{ d}}t):\,t\in [0,T],\,z\in \mathbb R _0\}$ in (2.1) and (4.3). Let $F$ be a random vector in $\mathbb R ^d$, measurable with respect to the union $\fancyscript{F}^1_T\cup \fancyscript{F}^2_T$. We denote by $D_{\cdot }$ the Gaussian Malliavin derivative, and denote by $\delta$ the Gaussian Skorohod integral over $(0,T]$, that is, for a suitable smooth stochastic process $\{G_t:t\ge 0\}$ in $\mathbb R ^d$,

$$\begin{aligned} \delta (G_{\cdot }):=\int \limits _0^T G_s \delta W_s, \end{aligned}$$

where the multivariate integral is taken componentwise. Both the operators $D$ and $\delta$ are linked by the equality

$$\begin{aligned} \delta \left(\left\langle F, G_{\cdot }\right\rangle \right)= \left\langle \delta \left(G_{\cdot }\right),F\right\rangle -\int \limits _0^T\left\langle D_t F, G_t\right\rangle {\text{ d}}t, \end{aligned}$$

given the $\sigma$-field $\fancyscript{F}_T^2$. Moreover, taking the conditional expectation, we get the duality;

$$\begin{aligned} \mathbb E \left[\int \limits _0^T\left\langle D_t F,G_t\right\rangle {\text{ d}}t\,\Big |\fancyscript{F}^2_T\right] =\mathbb E \left[\left\langle \delta (G_{\cdot }),F\right\rangle \,\big |\fancyscript{F}^2_T\right]. \end{aligned}$$

(5.1)

Here, we have assumed that $F\in \mathbb D ^{1,2}$ and $G\in Dom(\delta )$.

We next give an integration by parts formula, in the one dimension framework for simplicity, which is a key tool to integrate the derivative in the following conditional expectation for $g\in C^1_b(\mathbb R ;\mathbb R )$,

$$\begin{aligned} \mathbb E \left[g^{\prime }(X)Z\,\big |\fancyscript{F}^2_T\right]=\mathbb E \left[g(X)H\,\big |\fancyscript{F}^2_T\right], \end{aligned}$$

where $X, Z$, and $H$ are suitable random variables. By the chain rule of the Malliavin derivative, we have that given the $\sigma$-field $\fancyscript{F}^2_T$ and for $t\in [0,T]$,

$$\begin{aligned} \left(D_t g(X)\right) Z h_t=g^{\prime }(X) Z h_t D_t X, \end{aligned}$$

where $\{h_t:t\in [0,T]\}$ is a suitable “smooth” stochastic process such that the terms appearing below are well defined. Integration over $[0,T]$ leads to

$$\begin{aligned} g^{\prime }(X)Z=\int \limits _0^T \left(D_t g(X)\right) \frac{Z h_t}{\int \limits _0^T h_s D_sX {\text{ d}}s}{\text{ d}}t. \end{aligned}$$

Using the duality (5.1), we get

$$\begin{aligned} \mathbb E \left[g^{\prime }(X)Z\,\big |\fancyscript{F}^2_T\right]&= \mathbb E \left[\int \limits _0^T D_t g(X) \frac{Z h_t}{\int \nolimits _0^T h_s D_sX{\text{ d}}s}{\text{ d}}t \,\Big |\fancyscript{F}^2_T\right]\nonumber \\&= \mathbb E \left[g(X) \delta \left(\frac{Z h_{\cdot }}{\int \nolimits _0^T h_s D_sX{\text{ d}}s}\right)\,\Big |\fancyscript{F}^2_T\right]. \end{aligned}$$

(5.2)

This identity is often called the (conditional) integration-by-parts formula.

Finally, we describe the transformation of the Skorohod integral $\delta W_t$ to the standard Ito integral ${\text{ d}}W_t$ on the Gaussian space (that is, again with the jump component $\{Z_t:\,t\in [0,T]\}$ frozen). In short, the transformation is carried out as either

$$\begin{aligned} \int \limits _0^T \int \limits _0^T f(t,s){\text{ d}}W_s\delta W_t=\int \limits _0^T \int \limits _0^t \frac{f(t,s)+f(s,t)}{2}{\text{ d}}W_s {\text{ d}}W_t, \end{aligned}$$

(5.3)

or

$$\begin{aligned}&\int \limits _0^T \int \limits _0^T \int \limits _0^s f(t,s,u){\text{ d}}W_u {\text{ d}}W_s\delta W_t\nonumber \\&\quad =\int \limits _0^T \int \limits _0^t \int \limits _0^s\frac{f(t,s,u)+f(t,u,s)+f(s,t,u)+f(s,u,t)+f(u,t,s)+f(u,s,t)}{3!}{\text{ d}}W_u {\text{ d}}W_s {\text{ d}}W_t,\nonumber \\ \end{aligned}$$

(5.4)

for suitable deterministic functions $f(t,s)$ and $f(t,s,u)$. In our context, the integral $\int _0^t (t-s)e^{-\lambda (t-s)}{\text{ d}}Z_s$ in the first component of $\nabla _{\theta }X_t(\theta )$ is treated as a deterministic function in $t$, as the Poisson component is independent of the Brownian component and is set to be frozen when performing the relevant operations. We refer the reader to Sect. 1.3 of Nualart [27] for details of this transformation.