Calibration for multivariate Lévy-driven Ornstein-Uhlenbeck processes with applications to weak subordination

Abstract

Consider a multivariate Lévy-driven Ornstein-Uhlenbeck process where the stationary distribution or background driving Lévy process is from a parametric family. We derive the likelihood function assuming that the innovation term is absolutely continuous. Two examples are studied in detail: the process where the stationary distribution or background driving Lévy process is given by a weak variance alpha-gamma process, which is a multivariate generalisation of the variance gamma process created using weak subordination. In the former case, we give an explicit representation of the background driving Lévy process, leading to an innovation term which is a discrete and continuous mixture, allowing for the exact simulation of the process, and a separate likelihood function. In the latter case, we show the innovation term is absolutely continuous. The results of a simulation study demonstrate that maximum likelihood numerically computed using Fourier inversion can be applied to accurately estimate the parameters in both cases.

A The Connection Between LDOUPs and Self-Decomposability

In this appendix, we review the well-established connection between LDOUPs and self-decomposability.

All self-decomposable distributions are infinitely divisible and there is a one-to-one correspondence between the stationary solutions of LDOUPs and self-decomposable distributions, which we summarise in the following lemma [see Sato (1999, Theorems 17.5 and 17.11)].

Lemma 5

Fix $\lambda >0$ and let $\mathbf{X}$ be the LDOUP given by (4) with BDLP $\mathbf{Z}\sim L^n(\varvec{\mu },\varSigma ,\mathcal{Z})$.

• (i) For all $\mathbf{Z}\sim L^n(\varvec{\mu },\varSigma ,\mathcal{Z})$ satisfying (6), there exists a $\mathbf{Y}\sim SD^n$ such that $\mathbf{X}$ has stationary distribution $\mathbf{Y}$.

• (ii) For all $\mathbf{Y}\sim SD^n$, there exists a $\mathbf{Z}\sim L^n(\varvec{\mu },\varSigma ,\mathcal{Z})$ satisfying (6), unique in law, such that $\mathbf{X}$ has stationary distribution $\mathbf{Y}$.

• (iii) If $\mathbf{Z}\sim L^n(\varvec{\mu },\varSigma ,\mathcal{Z})$ does not satisfy (6), then $\mathbf{X}$ has no stationary distribution.

Furthermore, it is possible to convert between the characteristic exponents of the stationary distribution $\mathbf{Y}$ and the BDLP $\mathbf{Z}$ using the next result [see Sato (1999, Theorem 17.5) for (i) and Masuda (2004, Lemma 2.5) for (ii)].

Lemma 6

Fix $\lambda >0$ and let $\mathbf{X}$ be the LDOUP given by (4).

• (i) Let $\mathbf{Z}\sim L^n(\varvec{\mu },\varSigma ,\mathcal{Z})$ satisfying (6) be the BDLP of $\mathbf{X}$, then the stationary distribution $\mathbf{Y}$ has characteristic exponent (19).

• (ii) Let $\mathbf{Y}\sim SD^n$ be the stationary distribution of $\mathbf{X}$. Suppose $\varPsi _{\mathbf{Y}}$ is differentiable for all $\varvec{\theta }\ne \mathbf{0}$ and $\langle \nabla _{\varvec{\theta }} \varPsi _{\mathbf{Y}} (\varvec{\theta }),\varvec{\theta }\rangle \rightarrow \mathbf{0}$ as $\varvec{\theta }\rightarrow \mathbf{0}$, then the BDLP $\mathbf{Z}$ has characteristic exponent

  $$\begin{aligned} \varPsi _\mathbf{Z}(\varvec{\theta }) = \langle \nabla _{\varvec{\theta }} \varPsi _{\mathbf{Y}} (\varvec{\theta }),\varvec{\theta }\rangle ,\quad \varvec{\theta }\in \mathbb {R}^n. \end{aligned}$$

The next lemma is about $\mathbf{Z}^*(\varDelta )$ defined in (7), it is implied by Sato and Yamazato (1983, Theorem 2.2) and Kac’s theorem. It explains why the observations of a LDOUP form a AR(1) process, and why the stationary solution of a LDOUP gives rise to self-decomposable distributions.

Lemma 7

Let $\varDelta >0$ and $\mathbf{Z}\sim L^n$. For $t_0=0,t_1=\varDelta , \dots ,t_m= m\varDelta$,

$$\begin{aligned} \Bigg ( {\int _{t_{k-1}}^{t_k} e^{\lambda s}\,\mathrm{d}\mathbf{Z}(\lambda s)}\Bigg )_{k=1,\dots ,m} \end{aligned}$$

is an iid sequence equal in distribution to $\mathbf{Z}^*(\varDelta )$, which has characteristic exponent (20).

Remark 15

Applying (4) at the times $t=t_0,t_1,\dots ,t_m$, we have

$$\begin{aligned} \mathbf{X}(t_{k})= b\mathbf{X}(t_{k-1})+\mathbf{Z}_{b}^{(k)},\quad k=1,\dots ,m, \end{aligned}$$

(41)

where $b=e^{-\lambda \varDelta }$ and $\mathbf{Z}_{b}^{(k)}=e^{-\lambda \varDelta } \int _{t_{k-1}}^{t_k} e^{\lambda s}\,\mathrm{d}\mathbf{Z}(\lambda s)$. Now by Lemma 7, $\mathbf{Z}_{b}^{(k)}{\mathop {=}\limits ^{D}}e^{-\lambda \varDelta }\mathbf{Z}^*(\varDelta )$, $k=1,\dots ,m$, are iid, and $\mathbf{X}(t_{k-1})$, being a function of $\mathbf{X}_0,\mathbf{Z}_{b}^{(1)},\dots , \mathbf{Z}_{b}^{(k-1)}$ only, is independent of $\mathbf{Z}_{b}^{(k)}$. $\square$

From (41), the observations $\mathbf{X}(0),\dots ,\mathbf{X}(t_m)$ follow an AR(1) process with innovation terms $\mathbf{Z}_{b}^{(k)}$, $k=1,\dots ,m$. With a minor abuse of terminology, we call $\mathbf{Z}^*(\varDelta )$ the innovation term. As noted in Barndorff-Nielsen et al. (1998, Sections 3 and 5), (41) satisfies (5) with stationary distribution $\mathbf{Y}{\mathop {=}\limits ^{D}}\mathbf{X}(t_k){\mathop {=}\limits ^{D}}\mathbf{X}(t_{k-1})$, $b=e^{-\lambda \varDelta }$ and $\mathbf{Z}_b=\mathbf{Z}_{b}^{(k)}$. This demonstrates the connection between the stationary distribution of a LDOUP and self-decomposability.