Parameter estimation for the Rosenblatt Ornstein–Uhlenbeck process with periodic mean

Abstract

We study the least squares estimator for the drift parameter of the Langevin stochastic equation driven by the Rosenblatt process. Using the techniques of the Malliavin calculus and the stochastic integration with respect to the Rosenblatt process, we analyze the consistency and the asymptotic distribution of this estimator. We also introduce alternative estimators, which can be simulated, and we study their asymptotic properties.

Appendix: The basics of the Malliavin calculus

Here we present the tools from Malliavin calculus needed throughout the paper. See Nualart (2006) or Nourdin and Peccati (2012) for more details.

1.1 Multiple Wiener–Itô integrals

Let $(B_{t})_{t\in [0, T]}$ be a Brownian motion defined on a probability space $\left( \Omega ,\mathcal {F},\mathbb {P}\right)$. For a deterministic function $h\in L^{2}\left( [0,T]\right)$, the Wiener integral $\int _{0}^{T} h\left( s\right) dB\left( s\right)$ is also denoted by B(h) . The inner product $\int _{0}^{T} f\left( s\right) g\left( s\right) ds$ will be denoted by $\left\langle f,g\right\rangle _{L^{2}\left( [0,T]\right) }$.

For every $q\ge 1$, ${\mathcal {H} }^{B}_{q}$ denotes the qth Wiener chaos of B, defined as the closed linear subspace of $L^{2}(\Omega )$ generated by the random variables $\{H_{q}(B(h)),h\in L ^{2}([0, T]),\Vert h\Vert _{L ^{2} ([0,T])}=1\}$ where $H_{q}$ is the qth Hermite polynomial $H_{q}(x)=\frac{(-1) ^ {q}}{q!} e ^ {-\frac{x^ {2}}{2}}\frac{d}{x^ {n}}(e ^ {\frac{ x ^ {2}}{2}}).$

The mapping ${I_{q}(h^{\otimes q}):}=q!H_{q}(B(h))$ can be extended to a linear isometry between $L^{2}_{s}(\mathbb {R}^{q})$ the space of symmetric square integrable functions of $[0,T]^{q}$ (equipped with the modified norm ${ \sqrt{q!}}\Vert .\Vert _{L^{2}([0,T]^{q})}$) and $\mathcal {H}^{B}_{q}$ . When $f \in L^{2}([0,T]^{q})$, the random variable ${ I_{q}(f)}$ can be interpreted as a multiple Wiener–Itô integral of f of order q w.r.t. B and in this case, we write :

$$\begin{aligned} I_{q}(f) = \int _{[0,T]^{q}} f(y_{1},\ldots ,y_{q}) dB _{y_{1}}\ldots dB_{y_{q}}. \end{aligned}$$

(35)

From the many properties of multiple Wiener–Itô integrals we recall now two that we will need in our study. The first one is the isometry property, which states that for every $f \in L^{2}_{s}([0,T]^{q})$, $g\in L^{2}_{s}([0,T]^{p})$, with $p,q \ge 1$, the following holds:

$$\begin{aligned} \mathbf {E}\left[ I_{q}(f)I_{p}(g)\right] = {\left\{ \begin{array}{ll} p! \times \big \langle f, g \big \rangle _{L^2([0,T]^{p})} \qquad &{}\text {if }\quad p=q,\\ 0 \qquad \qquad \qquad \quad \quad &{}\text {if }\quad p \ne q. \end{array}\right. } \end{aligned}$$

(36)

The second one is the hypercontractivity property which states that for $f \in L^{2}_{s}([0,T]^{q})$, $q\ge 1$, the multiple Wiener–Itô integral $I_{q}(f)$ satisfies a hypercontractivity property (equivalence in $\mathcal {H}^{B}_{q}$ of all $L^{p}(\Omega )$ norms for all $p\ge 2$), which implies that for any $F\in \oplus _{l=1}^{q}\mathcal {H}^{B}_{l}$ (i.e. in a fixed sum of Wiener chaoses), we have

$$\begin{aligned} \left( \mathbf {E}\big [|F|^{p}\big ]\right) ^{1/p}\leqslant c_{p,q}\left( \mathbf {E}\big [|F|^{2} \big ]\right) ^{1/2}\ \text{ for } \text{ any } p\ge 2. \end{aligned}$$

(37)

It should be noted that the constants $c_{p,q}$ above are known with some precision when F is a single chaos term: indeed, by Corollary 2.8.14 in Nourdin and Peccati (2012), $c_{p,q}=\left( p-1\right) ^{q/2}$.

1.2 Malliavin derivative

Let $(B_{t})_{t \in [0, T]}$ be a Wiener process and let $\mathcal {S}$ be the class of smooth functionals of the form

$$\begin{aligned} F= f( B_{t_{1}},\ldots , B_{t_{n}}), \quad t_{1},\ldots , t_{n} \in [0, T], \end{aligned}$$

(38)

with $f \in C^{\infty } (\mathbb {R} ^{n})$ with at most polynomial growth (for f and its derivatives). For the random variable (38) we define its Malliavin derivative with respect to B by

$$\begin{aligned} D_{t}F= \sum _{i=1}^{n} \frac{\partial f}{\partial x_{i}} ( B_{t_{1}},\ldots , B_{t_{n}})1_{[0, t_{i}] } (t), t \in (0, T]. \end{aligned}$$

The operator D is an unbounded closable operator and it can be extended to the closure of $\mathcal {S}$ with respect to the norm

$$\begin{aligned} \Vert F\Vert _{k, p} ^{p} = \mathbf {E} \vert F \vert ^{p} +\sum _{j=1} ^{k} \mathbf {E} \Vert D ^{(j)} F \Vert ^{p} _{ L ^{2} ([0, T] ^{j})}, \quad F\in \mathcal {S}, p\ge 2, k\ge 1, \end{aligned}$$

denoted by $\mathbb D^{k,\,p}$.

We denote by $D^{(j)}$ the jth iterated Malliavin derivative. The Skorohod integral integral, denoted by $\delta$, is the adjoint operator of D. Its domain is

$$\begin{aligned} Dom (\delta )= \left\{ u \in L ^{2} \left( [0, T]\times \Omega \right) , \mathbf {E} \left| \int _{0} ^{T} u_{s} D_{s}F ds \right| \le C \Vert F\Vert _{2} \right\} \end{aligned}$$

and we have the duality relationship

$$\begin{aligned} \mathbf {E} F \delta (u)= \mathbf {E} \int _{0} ^{T} D_{s}F u_{s} ds , \quad F \in \mathcal {S}, u\in Dom (\delta ). \end{aligned}$$

We set $\mathbb {L} ^{k,p} = L^{p} ([0, T]; \mathbb {D} ^{k, p}), k\ge 1, p\ge 2$. This set is a subset of $\mathrm{Dom}(\delta )$.