Modern Stochastics: Theory and Applications

Consider the continuous-time linear model where ${B}^{H_{1}}$ and ${B}^{H_{2}}$ are two independent fractional Brownian motions with different Hurst indices $H_{1}$ and $H_{2}$ defined on some stochastic basis $(\varOmega ,\mathfrak{F},(\mathfrak{F})_{t},t\ge 0,\mathsf{P})$. We assume that the filtration is generated by these processes and completed by $\mathsf{P}$-negligible sets of $\mathfrak{F}_{0}$.

Recall that the fractional Brownian motion (fBm) ${B_{t}^{H}}$, $t\ge 0$, with Hurst index $H\in (0,1)$ is a centered Gaussian process with the covariance function

From now on we suppose that the Hurst indices in (1) satisfy the inequality and we consider the continuous modifications of both processes, which exist due to the Kolmogorov theorem. Assuming that the Hurst indices $H_{1}$, $H_{2}$ and parameters $\sigma _{1}\ge 0$, $\sigma _{2}\ge 0$ are known, we aim to estimate the unknown drift parameter θ by the continuous observations of the trajectories of X. Due to the long-range dependence property of fBm with $H>1/2$, we call our model the model with double long-range dependence.

In the case where $H_{1}=\frac{1}{2}$, the problem of drift parameter estimation in the model (1) was solved in [3], and in the case where $\frac{1}{2}<H_{1}<H_{2}<1$ and $H_{2}-H_{1}>1/4$, the estimator was constructed in [6]. The goal of the present paper is to generalize the results from [6] to arbitrary $\frac{1}{2}\le H_{1}<H_{2}<1$. The problem, more technical than principal, is that in the case where $H_{2}-H_{1}>1/4$ and $H_{1}>1/2$, the construction of the estimator is reduced to the question if the solution of the Fredholm integral equation of the 2nd kind with weakly singular kernel from $L_{2}[0,T]$ exists and is unique, but for $H_{2}-H_{1}\le 1/4$, the kernel does not belong to $L_{2}[0,T]$. Moreover, in this case, we can say that in the literature it is impossible to pick up for this kernel any suitable standard techniques for working with weak singular kernels, and it does not belong to any standard class of weak singular kernels. The matter lies in the fact that the kernel contains two power indices, $H_{1}$ and $H_{2}$, and they create more complex singularity than it usually happens. So, it is necessary to make many additional efforts in order to prove the compactness of the corresponding integral operator. Immediately after establishing the compactness of the corresponding integral operator, the problem of statistical estimation follows the same steps as in the paper [6], and we briefly present these steps for completeness.

The paper is organized as follows. In Section 2, we describe the model and explain how to reduce the solution of the estimation problem to the existence–uniqueness problem for the integral Fredholm equation of the 2nd kind with some nonstandard weakly singular kernel. In Section 3, we solve the existence–uniqueness problem. Section 4 is devoted to the basic properties of estimator, that is, we establish its form, consistency, and asymptotic normality. Section A contains the properties of hypergeometric function used in the proof of the existence–uniqueness result for the main Fredhom integral equation.

Since we suppose that the Hurst parameters $H_{1},H_{2}$ and scale parameters $\sigma _{1},\sigma _{2}$ are known, for technical simplicity, we consider the case where $\sigma _{1}=\sigma _{2}=1$ and, as it was mentioned before, $\frac{1}{2}\le H_{1}<H_{2}<1$. If we wish to include the unknown parameter θ into the fractional Brownian motion with the smallest Hurst parameter in order to apply Girsanov’s theorem for construction of the estimator, we consider a couple of processes $\{{\widetilde{B}}^{H_{1}}(t),{B}^{H_{2}}(t),\hspace{0.2778em}t\ge 0\}$, $i=1,2$, defined on the space $(\varOmega ,\mathfrak{F},(\mathfrak{F})_{t})$ and let $\mathsf{P}_{\theta }$ be a probability measure under which ${\widetilde{B}}^{H_{1}}$ and ${B}^{H_{2}}$ are independent, ${B}^{H_{2}}$ is a fractional Brownian motion with Hurst parameter $H_{2}$, and ${\widetilde{B}}^{H_{1}}$ is a fractional Brownian motion with Hurst parameter $H_{1}$ and drift θ, that is,

The probability measure $\mathsf{P}_{0}$ corresponds to the case $\theta =0$. Our main problem is the construction of maximum likelihood estimator for $\theta \in \mathbb{R}$ by the observations of the process $Z(t)=\theta t+{B}^{H_{1}}(t)+{B}^{H_{2}}(t)={\widetilde{B}}^{H_{1}}(t)+{B}^{H_{2}}(t),\hspace{0.2778em}t\in [0,T]$. As in [6], we apply to Z the linear transformation in order to reduce the construction to the sum with one term being the Wiener process. So, we take the kernel $l_{H}(t,s)={(t-s)}^{1/2-H}{s}^{1/2-H}$ and construct the integral where $\mathrm{B}(\alpha ,\beta )={\int _{0}^{1}}{x}^{\alpha -1}{(1-x)}^{\beta -1}dx$ is the beta function, and ${M}^{H_{1}}$ is a Gaussian martingale (Molchan martingale), admitting the representations with $\gamma _{H}={(2H(\frac{3}{2}-H)\varGamma {(3/2-H)}^{3}\varGamma (H+\frac{1}{2})\varGamma {(3-2H)}^{-1})}^{\frac{1}{2}}$ and a Wiener process W. According to [6], the linear transformation (2) is well defined, and the processes Z and Y are observed simultaneously. This means that we can reduce the original problem to the equivalent problem of the construction of maximum likelihood estimator of $\theta \in \mathbb{R}$ basing on the linear transformation Y. For simplicity, denote $\mathcal{B}_{H_{1}}:=\mathrm{B}(\frac{3}{2}-H_{1},\frac{3}{2}-H_{1})$. Now the main problem can be formulated as follows. Let $\frac{1}{2}\le H_{1}<H_{2}<1$, $i=1,2$, be a couple of processes defined on the space $(\varOmega ,\mathfrak{F})$, and $\mathsf{P}_{\theta }$ be a probability measure under which $\widetilde{X}_{1}$ and $X_{2}$ are independent, ${B}^{H_{2}}$ is a fractional Brownian motion with Hurst parameter $H_{2}$, and $\widetilde{X}_{1}$ is a martingale with square characteristics $\langle \widetilde{X}_{1}\rangle (t)=\frac{{\gamma _{H_{1}}^{2}}}{2-2H_{1}}{t}^{2-2H_{1}}$ and drift $\theta \mathcal{B}_{H_{1}}{t}^{2-2H_{1}}$, that is,

Also, denote $X_{1}(t)={M}^{H_{1}}(t)$. Our main problem is the construction of maximum likelihood estimator for $\theta \in \mathbb{R}$ by the observations of the process Note that, under the measure $\mathsf{P}_{\theta }$, the process is a Wiener process with drift. Denote $\delta _{H_{1}}=\frac{(2-2H_{1})\mathcal{B}_{H_{1}}}{\gamma _{H_{1}}}$.

By Girsanov’s theorem and independence of $X_{1}$ and $X_{2}$,

As it was mentioned in [3], the derivative of such a form is not the likelihood ratio for the problem at hand because it is not measurable with respect to the observed σ-algebra where $X(t)=X_{1}(t)+X_{2}(t)$.

We shall proceed as in [3]. Let $\mu _{\theta }$ be the probability measure induced by Y on the space of continuous functions with the supremum topology under probability $\mathsf{P}_{\theta }$. Then for any measurable set A, $\mu _{\theta }(A)=\int _{A}\varPhi (x)\mu _{0}(dx)$, where $\varPhi (x)$ is a measurable functional such that $\varPhi (X)=E_{0}(\frac{dP_{\theta }}{dP_{0}}|{\mathfrak{F}_{T}^{X}})$. This means that $\mu _{\theta }\ll \mu _{0}$ for any $\theta \in \mathbb{R}$. Taking into account that $\widetilde{X}_{1}=X_{1}$ under $\mathsf{P}_{0}$ and the fact that the vector process $(X_{1},X)$ is Gaussian, we get that the corresponding likelihood function is given by where $V(t)=\mathsf{E}_{0}{(X_{1}(t)-\mathsf{E}_{0}(X_{1}(t)|{\mathfrak{F}_{t}^{X}}))}^{2},\hspace{0.2778em}t\in [0,T]$.

The next reasonings repeat the corresponding part of [6]. We have to solve the following problem: to find the projection $P_{X}X_{1}(T)$ of $X_{1}(T)$ onto According to [4], the transformation formula for converting fBm into a Wiener process is of the form where $\beta _{H}={(\frac{H(2H-1)}{\mathrm{B}(H-1/2,2-2H)})}^{\frac{1}{2}}$, and the square-integrable kernel $K_{H}(t,s)$ is of the form

We have that $W_{i},\hspace{2.5pt}i=1,2$, are standard Wiener processes, which are obviously independent. Also, we have Then where For an interval $[0,T]$, denote by ${L_{H}^{2}}[0,T]$ the completion of the space of simple functions $f:[0,T]\to \mathbb{R}$ with respect to the scalar product where $\alpha _{H}=H(2H-1)$. Note that this space contains both functions and distributions. For functions from ${L_{H_{2}}^{2}}[0,T]$, we have that where

The projection of $X_{1}(T)$ onto $\{X(t),t\in [0,T]\}$ is a centered X-measurable Gaussian random variable and, therefore, is of the form with $h_{T}\in {L_{H_{1}}^{2}}[0,T]$. Note that $h_{T}$ still may be a distribution. However, as we will further see, it is a continuous function. The projection for all $u\in [0,T]$ must satisfy Using (5) together with independency of $X_{1}$ and $X_{2}$, we arrive at the equation where $\varepsilon _{H}={\gamma _{H}^{2}}/(2-2H)$. Finally, from (3)–(6) we get the prototype of a Fredholm integral equation where Differentiating (7), we get the Fredholm integral equation of the 2nd kind, where with the function $K_{H_{1},H_{2}}$ defined by (4).

We will establish in Remark 2 that for the case $H_{1}=\frac{1}{2}$, Eq. (8) can be reduced to the corresponding equation from [3]: but the difference between (10) and (8) lies in the fact that (10) can be characterized as the equation with standard kernel, whereas (8) with two different power exponents is more or less nonstandard, and, therefore, it requires an unconventional approach. On the one hand, it is known from the paper [6] that if the conditions $H_{2}-H_{1}>\frac{1}{4}$ and $H_{1}>1/2$ are satisfied, then Eq. (8) has a unique solution $h_{T_{n}}$ with $h_{T_{n}}(t){t}^{\frac{1}{2}-H_{1}}\in L_{2}[0,T_{n}]$ on any sequence of intervals $[0,T_{n}]$ except, possibly, a countable number of $T_{n}$ connected to eigenvalues of the corresponding integral operator (the meaning of this sentence will be specified later because, finally, we will get a similar result but in more general situation). On the other hand, the existence–uniqueness result for Eq. (10) in [3] is proved without any restriction on Hurst index $H_{2}$ while $H_{1}=\frac{1}{2}$. The difference between these results can be explained so that in [3] the authors state the existence and uniqueness of the continuous solution, whereas in [6] the solution is established in the framework of $L_{2}$-theory.

In this paper, we propose to consider Eq. (8) in the space $C[0,T]$ again. This means that we consider the corresponding integral operator as an operator from $C[0,T]$ into $C[0,T]$ and establish an existence–uniqueness result in $C[0,T]$. This approach has the advantage that we do not need anymore the assumption $H_{2}-H_{1}>\frac{1}{4}$ and can include the case $H_{1}=1/2$ again into the consideration.

We say that two integral equations are equivalent if they have the same continuous solutions. In this sense, Eqs. (7) and (8) are equivalent, and both are equivalent to the equation with continuous right-hand side, where

We get that the main problem (i.e., the MLE construction for the drift parameter) is reduced to the existence–uniqueness result for the integral equation (7).

Consider the integral operator K generated by the kernel K bearing in mind that the notations of the kernel and of the corresponding operator will always coincide:

Now we are in position to establish the properties of the kernel $\kappa (s,u)$ defined by (12) and (9). Introduce the notation ${[0,T]_{0}^{2}}={[0,T]}^{2}\setminus \{(0,0)\}$.

Now, consider the properties of the function participating in the kernel representation (13).

The following result establishes the way MLE for the drift parameter θ can be calculated. The proof of the theorem is the same as the proof of the corresponding statement from [6], so we omit it.

The next two results establish basic properties of the estimator; their proofs repeat the proofs of the corresponding statements from [6] and [3].