Parameter identification for the Hermite Ornstein–Uhlenbeck process

Abstract

By using the analysis on Wiener chaos, we study the behavior of the quadratic variations of the Hermite Ornstein–Uhlenbeck process, which is the solution to the Langevin equation driven by a Hermite process. We apply our results to the identification of the Hurst parameter of the Hermite Ornstein–Uhlenbeck process.

Appendix: Multiple stochastic integrals

Here, we shall only recall some elementary facts; our main reference is Nualart (2006). Consider ${\mathcal {H}}$ a real separable infinite-dimensional Hilbert space with its associated inner product ${\langle .,.\rangle }_{\mathcal {H}}$, and $(B (\varphi ), \varphi \in {\mathcal {H}})$ an isonormal Gaussian process on a probability space $(\Omega , {\mathfrak {F}}, \mathbb {P})$, which is a centered Gaussian family of random variables such that $\mathbf {E}\left( B(\varphi ) B(\psi ) \right) = {\langle \varphi , \psi \rangle }_{{\mathcal {H}}}$, for every $\varphi ,\psi \in {\mathcal {H}}$. Denote by $I_{q}$ the qth multiple stochastic integral with respect to B. This $I_{q}$ is actually an isometry between the Hilbert space ${\mathcal {H}}^{\odot q}$ (symmetric tensor product) equipped with the scaled norm $\frac{1}{\sqrt{q!}}\Vert \cdot \Vert _{{\mathcal {H}}^{\otimes q}}$ and the Wiener chaos of order q, which is defined as the closed linear span of the random variables $H_{q}(B(\varphi ))$ where $\varphi \in {\mathcal {H}},\;\Vert \varphi \Vert _{{\mathcal {H}}}=1$ and $H_{q}$ is the Hermite polynomial of degree $q\ge 1$ defined by:

$$\begin{aligned} H_{q}(x)=(-1)^{q} \exp \left( \frac{x^{2}}{2} \right) \frac{{\mathrm {d}}^{q} }{{\mathrm {d}x}^{q}}\left( \exp \left( -\frac{x^{2}}{2}\right) \right) ,\;x\in \mathbb {R}. \end{aligned}$$

(46)

The isometry of multiple integrals can be written as: for $p,\;q\ge 1$, $f\in {{\mathcal {H}}^{\otimes p}}$ and $g\in {{\mathcal {H}}^{\otimes q}}$,

$$\begin{aligned} \mathbf {E}\Big (I_{p}(f) I_{q}(g) \Big )= \left\{ \begin{array}{ll} q! \langle \tilde{f},\tilde{g} \rangle _{{\mathcal {H}}^{\otimes q}}&{}\quad \text{ if }\;p=q\\ 0 &{}\quad \text{ otherwise } \end{array}\right. \end{aligned}$$

(47)

where $\tilde{f}$ denotes the canonical symmetrization of f and it is defined by:

$$\begin{aligned} \tilde{f} (x_{1}, \ldots , x_{q}) =\frac{1}{q!}\sum _{\sigma \in \mathcal {S}_q} f(x_{\sigma (1) },\ldots , x_{\sigma (q)}), \end{aligned}$$

in which the sum runs over all permutations $\sigma$ of $\{1,\ldots ,q\}$. It also holds that:

$$\begin{aligned} I_{q}(f) = I_{q}\big ( \tilde{f}\big ). \end{aligned}$$

In the particular case when $\mathcal {H}=L^2(T, \mathcal {B}(T), \mu )$ , the rth contraction $f\otimes _{r}g$ is the element of ${\mathcal {H}}^{\otimes (p+q-2r)}$, which is defined by:

$$\begin{aligned}&(f\otimes _{r} g) ( s_{1}, \ldots , s_{p-r}, t_{1}, \ldots , t_{q-r}) \nonumber \\&=\int _{T ^{r} }\mathrm {d}u_{1}\ldots \mathrm {d}u_{r} f( s_{1}, \ldots , s_{p-r}, u_{1}, \ldots ,u_{r})g(t_{1}, \ldots , t_{q-r},u_{1}, \ldots ,u_{r}), \end{aligned}$$

(48)

for every $f\in L^2(T^p)$, $g\in L^2(T^q)$ and $r=1,\ldots ,p\wedge q$. The $f\tilde{\otimes }_{l}g$ we denote the symmetrization of the contraction $f\otimes _{l}g$.

The product for two multiple integrals can be expanded into a sum of multiple integrals [see Nualart (2006)]: if $f\in L^{2}(T^{n})$ and $g\in L^{2}(T^{m})$ are symmetric functions, then it holds that

$$\begin{aligned} I_{n}(f)I_{m}(g)=\sum _{l=0}^{m\wedge n}l!C_{m}^{l}C_{n}^{l}I_{m+n-2l}(f\otimes _{l}g). \end{aligned}$$

(49)

Another useful property of finite sums of multiple integrals is the hypercontractivity. Namely, if $F= \sum _{k=0} ^{n} I_{k}(f_{k})$ with $f_{k}\in \mathcal {H} ^{\otimes k}$ then

$$\begin{aligned} \mathbf {E}\vert F \vert ^{p} \le C_{p} \left( \mathbf {E}F ^{2} \right) ^{\frac{p}{2}}. \end{aligned}$$

(50)

for every $p\ge 2$.