Ergodicity and Drift Parameter Estimation for Infinite-Dimensional Fractional Ornstein–Uhlenbeck Process of the Second Kind

Abstract

We introduce the Hilbert-valued fractional Ornstein–Uhlenbeck of the second kind as the mild solution of a stochastic evolution equation with fractional-type Gaussian noise. We study the stationarity and the ergodicity for this infinite-dimensional process. Finally, via Malliavin calculus, we also analyze the least squares estimator of the drift parameter of the fractional Ornstein–Uhlenbeck of the second kind.

Appendix: Multiple Stochastic Integrals

In the present paragraph some elements of stochastic analysis are described that will be used in the paper. Let ${\mathcal {H}}$ be a real separable Hilbert space and $(B (\varphi ), \varphi \in {\mathcal {H}})$ an isonormal Gaussian process on a probability space $(\Omega , {\mathcal {A}}, P)$, that is a centered Gaussian family of random variables such that $\mathbf {E}\left( B(\varphi ) B(\psi ) \right) = \langle \varphi , \psi \rangle _{{\mathcal {H}}}$. Denote by $I_{n}$ the multiple stochastic integral with respect to B (see [21]). In fact, $I_{n}$ may be viewed as an isometry between the Hilbert space ${\mathcal {H}}^{\otimes n}$(symmetric tensor product) equipped with the scaled norm $\frac{1}{\sqrt{n!}}\Vert \cdot \Vert _{{\mathcal {H}}^{\otimes n}}$ and the Wiener chaos of order n which is defined as the closed linear span of the random variables $H_{n}(B(\varphi ))$ where $\varphi \in {\mathcal {H}}, \Vert \varphi \Vert _{{\mathcal {H}}}=1$ and $H_{n}$ is the Hermite polynomial of degree $n\ge 1$

$$\begin{aligned} H_{n}(x)=\frac{(-1)^{n}}{n!} \exp \left( \frac{x^{2}}{2} \right) \frac{d^{n}}{dx^{n}}\left( \exp \left( -\frac{x^{2}}{2}\right) \right) , \quad x\in \mathbb {R}. \end{aligned}$$

The isometry of multiple integrals can be written as follows: for m, n positive integers,

$$\begin{aligned} \mathbf {E}\left( I_{n}(f) I_{m}(g) \right)= & {} n! \langle f,g\rangle _{{\mathcal {H}}^{\otimes n}} \quad \hbox {if } m=n,\nonumber \\ \mathbf {E}\left( I_{n}(f) I_{m}(g) \right)= & {} 0 \quad \quad \quad \quad \qquad \;\;\hbox {if } m\not =n. \end{aligned}$$

(44)

Also,

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

holds, where $\tilde{f}$ denotes the symmetrization of f defined by

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

We recall the product formula for multiple integrals: if $f\in \mathcal {H} ^{\otimes n}$ and $g\in \mathcal {H} ^{\otimes n}$ are symmetric, then

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

(45)

where $f\otimes _{r}g$ denotes the contraction of order r ($0\le r\le m\wedge n$) of f and g (see [21]).

We denote by D the Malliavin derivative operator that acts on smooth functions of the form $F=g(B(\varphi _{1}), \ldots , B(\varphi _{n}))$, where g is a smooth function with compact support and $\varphi _{i} \in {\mathcal{{H}}}$,

$$\begin{aligned} DF=\sum _{i=1}^{n}\frac{\partial g}{\partial x_{i}}(B(\varphi _{1}), \ldots , B(\varphi _{n}))\varphi _{i}. \end{aligned}$$

By completion of the space of smooth functions with respect to suitable norms Sobolev-like spaces $\mathbb {D} ^{\alpha , p} \left( \mathcal{{H}}\right)$ may be introduced so that the (extended) operator D is continuous from $\mathbb {D} ^{\alpha , p}$ into $\mathbb {D} ^{\alpha -1, p} \left( \mathcal{{H}}\right)$ for $\alpha \ge 1$. In particular, if $F=I_{m}(f)$ with $f\in \mathcal {H} ^{\otimes m }$ is symmetric, then $D_{t}F= mI_{m-1} (\cdot , t)$.