Least-squares estimators based on the Adams method for stochastic differential equations with small Lévy noise

Abstract

We consider stochastic differential equations (SDEs) driven by small Lévy noise with some unknown parameters and propose a new type of least-squares estimators based on discrete samples from the SDEs. To approximate the increments of a process from the SDEs, we shall use not the usual Euler method but the Adams method, that is, a well-known numerical approximation of the solution to the ordinary differential equation appearing in the limit of the SDE. We show the consistency of the proposed estimators and the asymptotic distribution in a suitable observation scheme. We also show that our estimators can be better than the usual LSE based on the Euler method in the finite sample performance.

A Appendix

Lemma A.1

Let $\ell \in {\mathbb {N}}$. Let $\gamma _{\ell \nu }$ and $\beta _{\ell \nu }$ be given by (1.3) and (1.4). Then

$$\begin{aligned} \sum _{\nu =1}^{\ell } \left| \gamma _{\ell \nu } \right| \le \ell 2^{\ell -1} \quad (\ell =1,2,\dots ), \qquad \sum _{\nu =0}^{\ell } \left| \beta _{\ell \nu } \right| \le 2^{\ell } \quad (\ell =0,1,\dots ). \end{aligned}$$

(4.2)

Proof

The conclusion is obtained from

$$\begin{aligned} \sum _{\nu =1}^{\ell } \left| \gamma _{\ell \nu } \right|= & {} \sum _{\nu =1}^{\ell } \frac{1}{(\nu -1)!(\ell -\nu )!} \int _0^1 \prod _{\begin{array}{c} j=1 \\ j\ne \nu \end{array}}^\ell (u+j-1) \, \mathrm{d}u \\\le & {} \sum _{\nu =1}^{\ell } \frac{\ell !}{(\nu -1)!(\ell -\nu )!} = \ell 2^{\ell -1} \end{aligned}$$

for $\ell =1,2\dots$, and

$$\begin{aligned} \sum _{\nu =0}^{\ell } \left| \beta _{\ell \nu } \right| = \sum _{\nu =0}^{\ell } \frac{1}{\nu !(\ell -\nu )!} \int _0^1 \prod _{\begin{array}{c} j=0 \\ j\ne \nu \end{array}}^\ell (u+j-1) \, du \le \sum _{\nu =0}^{\ell } \frac{\ell !}{\nu !(\ell -\nu )!} = 2^{\ell } \end{aligned}$$

for $\ell =0,1,\dots$. $\square$

Lemma A.2

Let g be a continuous function on ${\mathbb {R}}^{d}$, let $t\mapsto y_t$ be an ${\mathbb {R}}^{d}$-valued continuous function on [0, 1], and let $\{f(\cdot ,\theta )\}_{\theta \in \Theta }$ be a pointwise equicontinuous family of functions from ${\mathbb {R}}^{d}$ to ${\mathbb {R}}^d$. If $\ell /n\rightarrow 0$ as $n\rightarrow \infty$, then

as $n\rightarrow \infty$, uniformly in $\theta \in \Theta$.

Proof

Since $\{f(y_\cdot ,\theta )\}_{\theta \in \Theta }$ is uniformly equicontinuous on [0, 1], for any $\eta >0$ there exists $N\in {\mathbb {N}}$, such that $\theta \in \Theta$, $|s-t|\le 1/N$ $\Rightarrow$ $\left| f(y_s,\theta ) - f(y_t,\theta ) \right| < \eta$. Then, for all $n\ge N$, $t\in [0,1)$ and $\theta \in \Theta$

and we have

uniformly in $(t,\theta )\in [0,1)\times \Theta$. By the continuity of g, we obtain

as $n\rightarrow \infty$, uniformly in $\theta \in \Theta$. Since $\{g\circ f(y_{\cdot },\theta )\}_{\theta \in \Theta }$ is equicontinuous at $t=0$, for $\ell \ge 2$

as $\ell /n\rightarrow 0$, uniformly in $\theta \in \Theta$. $\square$

Let $(\Omega ,P,{\mathscr {F}})$ be a probability space, and let ${{\,\mathrm{Sym}\,}}_{p}({\mathbb {R}})$ denote the set of all $p\times p$ symmetric matrix with real entries and with the Frobenius norm $\Vert \cdot \Vert _{F}$.

Lemma A.3

Suppose that $v_n\overset{p}{\rightarrow }v$ in ${\mathbb {R}}^p$ and $M_n\overset{p}{\rightarrow }M$ in ${{\,\mathrm{Sym}\,}}_p({\mathbb {R}})$ as $n\rightarrow \infty$, $w_n$ satisfies $v_n=M_n w_n$. If M is positive definite, $w_n\overset{p}{\rightarrow }M^{-1}v$.

Proof

Let $\eta$ be an arbitrary positive number less than the smallest eigenvalue of M. If $\Vert M_n-M\Vert _{F}<\eta$, then , where ${\mathbb {I}}_{p\times p}$ is the identity matrix of size p and is the Loewner order. This implies that $M_n$ is invetible and

Since $(M\pm \eta {\mathbb {I}}_{p\times p})^{-1}\rightarrow M^{-1}$ in ${{\,\mathrm{Sym}\,}}_{p}({\mathbb {R}})$ as $\eta \rightarrow 0$, there exists a positive number $\tilde{\eta }$ depending only on M, p and $\eta$, such that $\Vert M_n^{-1}-M^{-1}\Vert _F<\tilde{\eta }$ and $\tilde{\eta } \rightarrow 0$ as $\eta \rightarrow 0$.

Set ${\mathscr {D}}_n:=\{\omega \in \Omega \, | \, M_n(\omega )~\text {is invertible}\}$. Then, if an arbitrary positive number $\tilde{\eta }$ is sufficiently small, for some $\eta >0$, we have

$$\begin{aligned} P({\mathscr {D}}_n^\mathrm {C}) + P({\varvec{1}}_{{\mathscr {D}}_n}\Vert M_n^{-1}-M^{-1}\Vert _F>\tilde{\eta }) \le 2 P (\Vert M_n^{-1}-M^{-1}\Vert _F>\eta ) \rightarrow 0, \end{aligned}$$

where ${\varvec{1}}_A$ is the indicator function on a set $A\subset \Omega$. Hence, we obtain

$$\begin{aligned} w_n = M_n^{-1}v_n {\varvec{1}}_{{\mathscr {D}}_n} + w_n {\varvec{1}}_{{\mathscr {D}}_n^\mathrm {C}} \overset{p}{\rightarrow }M^{-1}v \end{aligned}$$

as $n\rightarrow \infty$. $\square$