Asymptotic Growth of Sample Paths of Tempered Fractional Brownian Motions, with Statistical Applications to Vasicek-Type Models

Abstract

Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) modify the power-law kernel in the moving average representation of fractional Brownian motion by introducing exponential tempering. We construct least-square estimators for the unknown drift parameters within Vasicek models that are driven by these processes. To demonstrate their strong consistency, we establish asymptotic bounds with probability 1 for the rate of growth of trajectories of tempered fractional processes.

1. Introduction

A number of recent papers have considered two broad classes of continuous stochastic Gaussian processes: tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII). They were presented in [1,2], respectively. These processes modify the power-law kernel in the moving average representation of a fractional Brownian motion (FBM) by incorporating exponential tempering. The paper [1] develops the basic theory of TFBM, including moving average and spectral representations, sample path properties, and an application for modeling wind speed. Sabzikar and Surgailis [2] introduced TFBMII, another tempered version of FBM, and discussed its basic properties, such as spectral representation and the covariance function and local and global asymptotic self-similarity. In particular, they established that TFBM and TFBMII are distinct processes. Notably, the large-time properties of TFBMII closely resemble those of FBM, presenting a stark contrast to the large-time properties of TFBM. More general tempered fractional stable motions of the first and the second kind were studied in [2,3], respectively, where the basic theory of tempered fractional stable processes was developed, including dependence structure, sample path behavior, local times, and local nondeterminism.

In contrast to FBM, TFBM and TFBMII can be defined for any value of the Hurst parameter $H>0$, attracting attention across various research fields. TFBM and TFBMII serve as valuable stochastic models in situations where data exhibit FBM characteristics at an intermediate scale but deviate from FBM at longer scales. In particular, both processes find application as models in turbulence and other applied areas.

An application of tempered fractional Brownian motion to geophysical flows is presented in [4], where a tempered fractional time series model for turbulence is proposed. The authors seek to enhance the modeling of turbulence in geophysical contexts, demonstrating the practical applicability of tempered fractional processes. The papers [5,6] focus on wavelet estimation and modeling of geophysical flows using tempered fractional Brownian motion. Concentrating on bifurcation dynamics, the paper [7] investigates the behavior of the tempered fractional Langevin equation. The authors analyze bifurcations, offering insights into the intricate dynamics that can emerge in systems described by tempered fractional processes. Extending the study beyond TFBM, the paper by [8] introduces a generalized Langevin equation with a tempered memory kernel. The authors explore the impact of tempered memory on the Langevin equation, contributing to the understanding of complex dynamical systems.

The Vasicek model, originally introduced by Vasicek [9] for the purpose of interest rate modeling, serves as an extension of the Langevin equation. This extension consisted of inserting the Brownian component as the additive noise. Presently, its applications extend beyond the realms of financial mathematics and economics. Furthermore, various extensions have been developed to broaden the Vasicek model’s specifications. One notable extension, motivated by the observed long-range dependence in hydrology, geophysics, climatology, telecommunication, economics, and finance data, involves replacing the Brownian motion in the Vasicek model with an FBM $B^H$. This modification leads to the fractional Vasicek model, represented by the stochastic differential equation:

$$d{Y}_t=(a+b{Y}_t)dt+\sigma d{B}_t^H,t\ge 0.$$

(1)

Over the past decade, numerous studies have emerged focusing on the statistical inference within this model. In particular, investigations into the least squares and ergodic-type estimators of the unknown parameters a and b have been conducted by Xiao and Yu [10,11]. The maximum likelihood estimators for parameters a and b were explored in [12]. Additionally, several researchers have delved into statistical inference within the Model (1), introducing different types of Gaussian noises. For instance, Prakasa Rao considered maximum likelihood estimation for sub-fractional [13] and mixed fractional [14] Vasicek models. In [15], the authors investigated least-squares estimation in the Vasicek model driven by a Gaussian process belonging to a wide class, including FBM, bifractional, and sub-fractional Brownian motions (see also [16] for the application of these results to other examples of noise).

Recently, Azmoodeh et al. [17] conducted an investigation into the effects of tempering on various local and global properties of FBM. Specifically, their study revealed that tempering does not alter the local properties of FBM, including sample paths and p-variation. However, it has a substantial impact on the Breuer–Major theorem, the asymptotic behavior of the third and fourth cumulants of FBM, and the optimal fourth-moment theorem. Furthermore, in [17], the asymptotic properties of the variance of tempered fractional processes for large time t were studied. The authors established that, in contrast to FBM, the variance of TFBM converges to a finite constant as time approaches infinity. Conversely, they demonstrated that the variance of TFBMII is proportional to $ct$ for large t.

The primary objective of this paper is to delve more deeply into the properties of trajectories of tempered fractional processes. Specifically, our goal is to investigate the asymptotic growth (as time increases to ∞) with probability 1 for the trajectories of TFBM and TFBMII. In order to achieve this goal, we study in detail the behavior of these processes and their covariance functions at various time intervals. More precisely, we start with an exploration of the behavior of variances of tempered fractional processes as time approaches zero. Then, combining the knowledge of this behavior with the large-time asymptotic properties obtained in [17], we construct upper bounds for the variances on the whole positive half-axis. These bounds imply Hölder properties of TFBM and TFBMII via the Kolmogorov–Čentsov theorem. Moreover, these results enable us to establish asymptotic growth of the trajectories of both processes using the general theorems for Gaussian processes, developed in [18,19] (see Appendix B for details). As an application of our results, we consider a statistical problem of drift parameter estimation in the Vasicek model driven by either TFBM or TFBMII and prove the strong consistency of least-squares-type estimators. We also discuss a generalization of the developed statistical methods to the Vasicek model with processes exhibiting similar (power) asymptotic growth, such as multifractional Brownian motion and Gaussian Volterra processes with power-type kernels.

The paper is structured as follows. Section 2 provides a review of the definitions and fundamental properties of TFBM and TFBMII. In Section 3, we analyze the asymptotic behavior of the variances for both processes at zero. As corollaries, we establish upper bounds for the incremental variances of TFBM and TFBMII. These bounds enable an exploration of their sample path properties. Section 4 introduces almost sure upper bounds for the trajectories of TFBM and TFBMII. The statistical application of these bounds to estimate drift parameters in the Vasicek model, driven by tempered processes, is discussed in Section 5. Some technical proofs are given in Appendix A. For the reader’s convenience, Appendix B contains the theory regarding the asymptotic growth of centered Gaussian processes, as developed in [18]. This theory is the principal tool in establishing the main results of Section 4.

2. Preliminaries of the Tempered Processes

Let $W=\left\{W_s,s\in \mathbb{R}\right\}$ be a two-sided Wiener process, $H>0$, $\lambda >0$.

Definition 1.

The stochastic process $B_{H,\lambda }^I={\left\{B_{H,\lambda }^I\left(t\right)\right\}}_{t\in \mathbb{R}}$ defined by the Wiener integral

$$B_{H,\lambda }^I\left(t\right):={\int }_{\mathbb{R}}\left[g_{H,\lambda ,t}^I\left(s\right)\right]d{W}_s,$$

(2)

where

$$g_{H,\lambda ,t}^I\left(s\right):=e^{-\lambda {(t-s)}_{+}}{(t-s)}_{+}^{H-\frac{1}{2}}-e^{-\lambda {(-s)}_{+}}{(-s)}_{+}^{H-\frac{1}{2}},s\in \mathbb{R},$$

is called a tempered fractional Brownian motion (TFBM).

Definition 2.

The stochastic process $B_{H,\lambda }^{II}={\left\{B_{H,\lambda }^{II}\left(t\right)\right\}}_{t\in \mathbb{R}}$ defined by the Wiener integral

$$B_{H,\lambda }^{II}\left(t\right):={\int }_{\mathbb{R}}{g}_{H,\lambda ,t}^{II}\left(s\right)d{W}_s,$$

(3)

where

$$\begin{array}{cc}\hfill g_{H,\lambda ,t}^{II}\left(s\right)& :=e^{-\lambda {(t-s)}_{+}}{(t-s)}_{+}^{H-\frac{1}{2}}-e^{-\lambda {(-s)}_{+}}{(-s)}_{+}^{H-\frac{1}{2}}\hfill \\ \hfill & +\lambda {\int }_0^t{(u-s)}_{+}^{H-\frac{1}{2}}{e}^{-\lambda {(u-s)}_{+}}du,s\in \mathbb{R},\hfill \end{array}$$

is called a tempered fractional Brownian motion of the second kind (TFBMII).

Next, we recall the basic properties of TFBM (Formula (2)) and TFBMII (Formula (3)), respectively. For details, we refer the reader to [1,2].

Proposition 1. 

1.

TFBM (2) and TFBMII (3) are Gaussian stochastic processes with stationary increments, having the following scaling property: for any scaling factor $c>0$

$${\left\{X_{H,\lambda }\left(ct\right)\right\}}_{t\in \mathbb{R}}\stackrel{▵}{=}{\left\{c^H{X}_{H,c\lambda }\left(t\right)\right\}}_{t\in \mathbb{R}},$$

(4)

where $X_{H,\lambda }$ could be $B_{H,\lambda }^I$ or $B_{H,\lambda }^{II}$ (here $\stackrel{▵}{=}$ denotes the equality of all finite-dimensional distributions).

2.

From (4)

$$\mathbf{E}\left[(X_{H,\lambda }{\left(\right|t\left|\right))}^2\right]={\left|t\right|}^{2H}\mathbf{E}\left[{\left(X_{H,\lambda \left|t\right|}\left(1\right)\right)}^2\right]:={\left|t\right|}^{2H}{C}_t^2,$$

(5)

where the function $C_t^2$ has the following explicit representation.

(i)

If $X_{H,\lambda }=B_{H,\lambda }^I$, then

$$C_t^2={\left(C_t^I\right)}^2=\mathbf{E}\left[{\left(B_{H,\lambda \left|t\right|}^I\left(1\right)\right)}^2\right]=\frac{2\mathsf{\Gamma }\left(2H\right)}{{\left(2\lambda \right|t\left|\right)}^{2H}}-\frac{2\mathsf{\Gamma }(H+\frac{1}{2})}{\sqrt{\pi }}\frac{1}{{\left(2\lambda \right|t\left|\right)}^H}{K}_H\left(\lambda \right|t\left|\right),$$

(6)

where $t\ne 0$, and $K_{\nu }\left(z\right)$ is the modified Bessel function of the second kind.

(ii)

If $X_{H,\lambda }=B_{H,\lambda }^{II}$, then

$$\begin{array}{cc}\hfill C_t^2& ={\left(C_t^{II}\right)}^2=\mathbf{E}\left[(B_{H,\lambda \left|t\right|}^{II}{\left(\right|1\left|\right))}^2\right]=\frac{(1-2H)\mathsf{\Gamma }(H+\frac{1}{2})\mathsf{\Gamma }\left(H\right){\left(\lambda t\right)}^{-2H}}{\sqrt{\pi }}\hfill \\ \hfill & \times \left[1-{}_{2}F_3\left(\{1,-1/2\},\{1-H,1/2,1\},{\lambda }^2{t}^2/4\right)\right]\hfill \\ \hfill & +\frac{\mathsf{\Gamma }(1-H)\mathsf{\Gamma }(H+\frac{1}{2})}{\sqrt{\pi }H{2}^{2H}}{}_{2}F_3\left(\{1,H-1/2\},\{1,H+1,H+1/2\},{\lambda }^2{t}^2/4\right),\hfill \end{array}$$

(7)

where ${}_{2}F_3$ is the generalized hypergeometric function.

3.

Covariance functions of tempered processes have the following form.

(i)

TFBM (2) with parameters $H>0$ and $\lambda >0$ has the covariance function

$$R(t,s)=Cov[B_{H,\lambda }^I\left(t\right),B_{H,\lambda }^I\left(s\right)]=\frac{1}{2}\left[C_t^2{\left|t\right|}^{2H}+C_s^2{\left|s\right|}^{2H}-C_{t-s}^2{|t-s|}^{2H}\right],$$

for any $t,s\in \mathbb{R}$, where $C_t^2={\left(C_t^I\right)}^2$ is given by (6).

(ii)

TFBMII (3) with parameters $H>0$ and $\lambda >0$ has the covariance function

$$R^{II}(t,s)=Cov[B_{H,\lambda }^{II}\left(t\right),B_{H,\lambda }^{II}\left(s\right)]=\frac{1}{2}\left[C_t^2{\left|t\right|}^{2H}+C_s^2{\left|s\right|}^{2H}-C_{t-s}^2{|t-s|}^{2H}\right],$$

(8)

for any $t,s\in \mathbb{R}$, where $C_t^2={\left(C_t^{II}\right)}^2$ is given by (7).

4.

The next two properties follow from ([17], Proposition 2.4).

(i)

The TFBM (2) with parameters $H>0$ and $\lambda >0$ satisfies

$$\underset{t\to +\infty }{lim}\mathbf{E}{\left[B_{H,\lambda }^I\left(t\right)\right]}^2=\frac{2\mathsf{\Gamma }\left(2H\right)}{{\left(2\lambda \right)}^{2H}}.$$

(9)

(ii)

The TFBMII (3) with parameters $H>0$ and $\lambda >0$ satisfies

$$\underset{t\to +\infty }{lim}\mathbf{E}{\left[\frac{B_{H,\lambda }^{II}\left(t\right)}{\sqrt{t}}\right]}^2={\lambda }^{1-2H}{\mathsf{\Gamma }}^2\left(H+\frac{1}{2}\right).$$

(10)

Now, we recall the following result from ([17], Theorem 2.7), which states that TFBM and TFBMII are quasi-helices, in the geometric terminology of Kahane [20].

Theorem 1.

Let X stand for a TFBM $B_{H,\lambda }^I$ from (2) or for a TFBMII $B_{H,\lambda }^{II}$ from (3) both with $0<H<1$ and $\lambda >0$. Then, there exist positive constants $C_1$ and $C_2$ such that

$$C_1{|t-s|}^{2H}\le {\mathbf{E}\left[\right|X\left(t\right)-X\left(s\right)|}^2]\le C_2{|t-s|}^{2H},$$

(11)

for any $s,t\in [0,1]$.

3. Asymptotic Behavior of Variances of TFBM and TFBMII at Zero and Upper Bounds for the Incremental Variances

In order to examine the behavior of trajectories in detail, we analyze them not only at infinity but also at zero. Interesting effects arise when $H=1$, and this is a consequence of tempering because, without tempering, fractional Brownian motion with $H=1$ is simply a linear process of the form $\xi t$, $t\ge 0$, where $\xi$ is a standard Gaussian random variable.

We start with an investigation of the asymptotic behavior of the variances of TFBM and TFBMII at zero. Recall that their behavior at infinity was established in ([17], Proposition 2.4), and it is given by the Formulas (9) and (10).

The knowledge of the behavior of the variances for small and for large t enables us to extend the results of Theorem 1. Namely, we shall establish upper bounds for incremental variances for arbitrary $H>0$ and arbitrary times $t,s\in {\mathbb{R}}_{+}$.

3.1. Asymptotic Behavior of Variance of TFBM at Zero

The following lemma describes the behavior of the variance of TFBM at zero for various values of H. The proof of this result is provided in Appendix A.

Lemma 1. 

(i)

For any $H\in (0,1)$

$$\mathbf{E}\left[B_{H,\lambda }^I{\left(t\right)}^2\right]\sim \frac{\mathsf{\Gamma }{(H+\frac{1}{2})}^2}{2Hsin\left(\pi H\right)\mathsf{\Gamma }\left(2H\right)}{t}^{2H},\mathrm{as}t\downarrow 0.$$

(12)

(ii)

For $H=1$

$$\mathbf{E}\left[B_{1,\lambda }^I{\left(t\right)}^2\right]\sim -\frac{t^2}{4}logt,\mathrm{as}t\downarrow 0.$$

(iii)

For any $H>1$

$$\mathbf{E}\left[B_{H,\lambda }^I{\left(t\right)}^2\right]\sim \frac{\mathsf{\Gamma }\left(2H\right)}{2^{2H+1}{\lambda }^{2H-2}(H-1)}{t}^2,\mathrm{as}t\downarrow 0.$$

Remark 1.

Let $H\in (0,1)$. When $\lambda =0$, we obtain (up to a multiplicative constant) the conventional (untempered) fractional Brownian motion (FBM) through the Mandelbrodt–van Ness representation:

$$B_H\left(t\right):={\int }_{\mathbb{R}}\left[{(t-s)}_{+}^{H-\frac{1}{2}}-{(-s)}_{+}^{H-\frac{1}{2}}\right]d{W}_s.$$

(13)

It is known (see, e.g., [21], Theorem 1.3.1) that this process has the variance

$$\mathbf{E}\left[{\left(B_H\left(t\right)\right)}^2\right]=\frac{\mathsf{\Gamma }{(H+\frac{1}{2})}^2}{2Hsin\left(\pi H\right)\mathsf{\Gamma }\left(2H\right)}·t^{2H},t\ge 0.$$

Hence, the relation (12) means that the variance of TFBM is asymptotically equivalent to the variance of FBM as $t\downarrow 0$.

Note that in the case $H\ge 1$, the integral in (13) does not exist. The FBM is defined for $H=1$ by a different approach. This explains the difference in the behavior of TFBM and FBM in the case $H=1$.

Remark 2.

Using the harmonizable representation of TFBM, Meerschaert and Sabzikar [1] established that for $H\in (0,1)$

$$sup\left\{\beta >0:\mathbf{E}\left[B_{H,\lambda }^I{\left(t\right)}^2\right]=o\left(t^{2\beta }\right)\mathrm{as}t\downarrow 0\right\}=2H,$$

see the proof of ([1], Theorem 5.1). In Lemma 1, we specify this result, establishing the precise asymptotic equivalence as $t\downarrow 0$. Moreover, we also study the behavior of TFBM at zero for $H=1$ and $H>1$.

Lemma 2.

(i)

For any $H\in (0,1)$ and for all $t,s\in {\mathbb{R}}_{+}$

$$\mathbf{E}\left[|B_{H,\lambda }^I\left(t\right)-B_{H,\lambda }^I\left(s\right){|}^2\right]\le C\left({\left|t-s\right|}^{2H}\wedge 1\right).$$

(ii)

If $H=1$, then for all $t,s\in {\mathbb{R}}_{+}$

$$\mathbf{E}\left[|B_{H,\lambda }^I\left(t\right)-B_{H,\lambda }^I\left(s\right){|}^2\right]\le C\left({\left|t-s\right|}^2\left|log\left|t-s\right|\right|\wedge 1\right).$$

(iii)

For any $H>1$ and for all $t,s\in {\mathbb{R}}_{+}$

$$\mathbf{E}\left[|B_{H,\lambda }^I\left(t\right)-B_{H,\lambda }^I\left(s\right){|}^2\right]\le {C\left(\right|t-s|}^2\wedge 1).$$

Proof. 

The proof follows from Lemma 1 and Proposition 1. Indeed, the convergence (9) implies that

$$\mathbf{E}\left[|B_{H,\lambda }^I\left(t\right)-B_{H,\lambda }^I\left(s\right){|}^2\right]\le C,$$

(14)

for some $C>0$, any $H>0$ and for all $t,s\in {\mathbb{R}}_{+}$. Further, for $H\in (0,1)$ it follows from (12) and (9) that

$$\frac{\mathbf{E}\left[B_{H,\lambda }^I{\left(t\right)}^2\right]}{t^{2H}}\to \frac{\mathsf{\Gamma }{(H+\frac{1}{2})}^2}{2Hsin\left(\pi H\right)\mathsf{\Gamma }\left(2H\right)},\mathrm{as}t\downarrow 0,\mathrm{and}\frac{\mathbf{E}\left[B_{H,\lambda }^I{\left(t\right)}^2\right]}{t^{2H}}\to 0,\mathrm{as}t\to \infty ,$$

whence in view of the continuity of the function $\mathbf{E}\left[B_{H,\lambda }^I{\left(t\right)}^2\right]/t^{2H}={\left(C_t^I\right)}^2$, we obtain

$$\frac{\mathbf{E}\left[B_{H,\lambda }^I{\left(t\right)}^2\right]}{t^{2H}}\le C\mathrm{for}\mathrm{all}t,s\in {\mathbb{R}}_{+}.$$

Taking into account that with Proposition 1,

$$\mathbf{E}\left[|B_{H,\lambda }^I\left(t\right)-B_{H,\lambda }^I\left(s\right){|}^2\right]={\left(C_{\left|t-s\right|}^I\right)}^2{\left|t-s\right|}^{2H}=\mathbf{E}\left[B_{H,\lambda }^I{(t-s)}^2\right],$$

we obtain the statement $\left(i\right)$. The statements $\left(ii\right)$ and $\left(iii\right)$ are derived from the corresponding statements of Lemma 1 using a similar argument.  □

Remark 3.

Meerschaert and Sabzikar [1] have obtained the following results about the sample path properties of TFBM: (a) If $H\in (0,1)$, then TFBM satisfies the Hölder condition of any order $\gamma \in (0,H)$ on any interval $[0,T]$ ([1], Theorem 5.1); (b) For $H>1$, TFBM has continuously differentiable sample paths and, moreover, they are p times continuously differentiable for $H>p$ ([1], Remark 5.2). Note that (a) agrees with statement $\left(i\right)$ of our Lemma 2, and follows from it via the Kolmogorov–Čentsov theorem.

In the case $H=1$, it follows from statement $\left(ii\right)$ of Lemma 2 that for any $ϵ>0$, there exists $C_{ϵ}>0$, such that $\mathbf{E}|B_{H,\lambda }^I\left(t\right)-B_{H,\lambda }^I\left(s\right){|}^2\le C_{ϵ}{\left|t-s\right|}^{2-ϵ}$. Then, according to ([22], Theorem 1), for $H=1$, TFBM satisfies the Hölder condition of any order $\gamma \in (0,1)$.

In order to investigate differentiability (for $H\ge 1$), let us write $B_{H,\lambda }^I\left(t\right)=B_{H,\lambda ,-}^I\left(t\right)+B_{H,\lambda ,+}^I\left(t\right)$, where

$$\begin{array}{c}{B}_{H,\lambda ,-}^I\left(t\right)={\int }_{-\infty }^0\left(e^{-\lambda (t-s)}{(t-s)}^{H-\frac{1}{2}}-e^{-\lambda (-s)}{(-s)}^{H-\frac{1}{2}}\right)d{W}_s,\\ B_{H,\lambda ,+}^I\left(t\right)={\int }_0^t{e}^{-\lambda (t-s)}{(t-s)}^{H-\frac{1}{2}}d{W}_s.\end{array}$$

We study each of the two terms separately. Denoting ${\tilde{W}}_s=W_{-s}$ and integrating by parts, we obtain

$$\begin{array}{cc}\hfill B_{H,\lambda ,-}^I\left(t\right)& ={\int }_0^{\infty }{e}^{-\lambda s}\left(e^{-\lambda t}{(t+s)}^{H-\frac{1}{2}}-s^{H-\frac{1}{2}}\right)d{\tilde{W}}_s\hfill \\ \hfill & ={\int }_0^{\infty }{\tilde{W}}_s\left[-\lambda e^{-\lambda s}\left(e^{-\lambda t}{(t+s)}^{H-\frac{1}{2}}-s^{H-\frac{1}{2}}\right)\right.\hfill \\ \hfill & +\left.\left(H-\frac{1}{2}\right)e^{-\lambda s}\left(e^{-\lambda t}{(t+s)}^{H-\frac{3}{2}}-s^{H-\frac{3}{2}}\right)\right]ds.\hfill \end{array}$$

The derivative of this process is well-defined for $H>1$, and it equals

$$\begin{array}{c}\frac{d}{dt}{B}_{H,\lambda ,-}^I\left(t\right)=e^{-\lambda t}{\int }_0^{\infty }{\tilde{W}}_s{e}^{-\lambda s}\left[{\lambda }^2{(t+s)}^{H-\frac{1}{2}}-2\lambda \left(H-\frac{1}{2}\right){(t+s)}^{H-\frac{3}{2}}\right.\hfill \\ \hfill +\left.{\left(H-\frac{1}{2}\right)}^2{(t+s)}^{H-\frac{5}{2}}\right]ds,\end{array}$$

since the last integral converges a. s. for any $t\ge 0$. Indeed, the Wiener process $\tilde{W}$ is a Hölder continuous of order $\frac{1}{2}-ϵ$ on any $[0,T]$ for any $ϵ\in (0,\frac{1}{2})$, whence $\left|{\tilde{W}}_s\right|\le C_T\left(\omega \right)s^{\frac{1}{2}-ϵ}\le C_T\left(\omega \right){(t+s)}^{\frac{1}{2}-ϵ}$ and

$$\begin{array}{c}\left|\frac{d}{dt}{B}_{H,\lambda ,-}^I\left(t\right)\right|\le {\int }_0^{\infty }{e}^{-\lambda s}\left[{\lambda }^2{(t+s)}^{H-ϵ}+2\lambda \left(H-\frac{1}{2}\right){(t+s)}^{H-1-ϵ}\right.\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left.{\left(H-\frac{1}{2}\right)}^2{(t+s)}^{H-2-ϵ}\right]ds<\infty \hfill \end{array}$$

a.s. for any $H>1$ and any $t\in [0,T]$, since we can choose $ϵ<H-1$ in order to guarantee the convergence of the integral for $t=0$. Note that for $H=1$, the above integral converges for any $t>0$, but it diverges for $t=0$ because in this case

$$\begin{array}{cc}\hfill \mathbf{E}{\left({\int }_0^{\infty }{\tilde{W}}_s{e}^{-\lambda s}{s}^{H-\frac{5}{2}}ds\right)}^2& =\mathbf{E}{\left({\int }_0^{\infty }{\tilde{W}}_s{e}^{-\lambda s}{s}^{-\frac{3}{2}}ds\right)}^2\hfill \\ \hfill & ={\int }_0^{\infty }{\int }_0^{\infty }(s\wedge u)e^{-\lambda s-\lambda u}{s}^{-\frac{3}{2}}{u}^{-\frac{3}{2}}duds,\hfill \end{array}$$

and

$${\int }_0^1{\int }_0^s(s\wedge u)e^{-\lambda s-\lambda u}{s}^{-\frac{3}{2}}{u}^{-\frac{3}{2}}dyds\ge e^{-2\lambda }{\int }_0^1{\int }_0^s{s}^{-\frac{3}{2}}{u}^{-\frac{1}{2}}duds=+\infty .$$

Similarly, one can see that for $H>p$ the process $B_{H,\lambda ,-}^I\left(t\right)$ is p times continuously differentiable with

$$\begin{array}{c}\frac{d^p}{d{t}^p}{B}_{H,\lambda ,-}^I\left(t\right)={\int }_0^{\infty }{\tilde{W}}_s{e}^{-\lambda s}\frac{d^{(p+1)}}{d{t}^{(p+1)}}\left[e^{-\lambda t}{(t+s)}^{H-\frac{1}{2}}\right]ds\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=e^{-\lambda t}{\int }_0^{\infty }{\tilde{W}}_s{e}^{-\lambda s}\sum _{k=0}^{p+1}\left(\genfrac{}{}{0pt}{}{p+1}{k}\right){(-\lambda )}^{p+1-k}{\left(H-\frac{1}{2}\right)}^k{(t+s)}^{H-\frac{1}{2}-k}ds.\hfill \end{array}$$

Now, let us investigate the differentiability of the integral $B_{H,\lambda ,+}^I\left(t\right)$. First, note that the process $M_t={\int }_0^t{e}^{\lambda s}d{W}_s$ is a.s. Hölder continuous of any order up to $\frac{1}{2}$. Further, assume that $H\in (p,p+1]$ for some $p\in \mathbb{N}$. Then the process

$${\int }_0^s{(s-u)}^{H-1/2-p}{e}^{\lambda u}d{W}_u={\int }_0^s{(s-u)}^{H-1/2-p}d{M}_u$$

(15)

is well-defined and Hölder continuous up to order $H-p$ according to ([23], Lemma 2.1). Taking into account the continuity of this integral, we conclude that the following integral exists with probability 1:

$$\begin{array}{c}{\int }_0^t{(t-s)}^{p-1}{\int }_0^s{(s-u)}^{H-1/2-p}{e}^{\lambda u}d{W}_{u}ds\hfill \\ ={\int }_0^t{e}^{\lambda u}{\int }_u^t{(t-s)}^{p-1}{(s-u)}^{H-1/2-p}dsd{W}_u\\ =B\left(p,H+\frac{1}{2}-p\right){\int }_0^t{e}^{\lambda u}{(t-u)}^{H-1/2}d{W}_u\\ =\frac{(p-1)!\mathsf{\Gamma }(H+\frac{1}{2}-p)}{\mathsf{\Gamma }(H+\frac{1}{2})}{e}^{\lambda t}{B}_{H,\lambda ,+}^I\left(t\right).\end{array}$$

Hence,

$$B_{H,\lambda ,+}^I\left(t\right)=\frac{\mathsf{\Gamma }(H+\frac{1}{2})}{(p-1)!\mathsf{\Gamma }(H+\frac{1}{2}-p)}{e}^{-\lambda t}{\int }_0^t{(t-s)}^{p-1}{\int }_0^s{(s-u)}^{H-1/2-p}{e}^{\lambda u}d{W}_{u}ds.$$

We see that this process is p times differentiable: the pth derivative in variable t of the integral ${\int }_0^t{(t-s)}^{p-1}{\int }_0^s{(s-u)}^{H-1/2-p}{e}^{\lambda u}d{W}_{u}ds$ is equal to $(p-1)!{\int }_0^t{(t-u)}^{H-1/2-p}{e}^{\lambda u}d{W}_u$.

Note that in the case $H=1$ the integral (15) does not exist even for $p=1$ since the function $u\mapsto {(s-u)}^{-1/2}{e}^{\lambda u}$ is not square-integrable on $[0,s]$. Moreover, it turns out that $B_{1,\lambda ,+}^I\left(t\right)$ is non-differentiable for any $t>0$. To show this, let us consider

$$\begin{array}{cc}\hfill B_{1,\lambda ,+}^I\left(t\right)& ={\int }_0^t{e}^{-\lambda (t-s)}{(t-s)}^{\frac{1}{2}}d{W}_s=-{\int }_0^t{W}_{s}d\left(e^{-\lambda (t-s)}{(t-s)}^{\frac{1}{2}}\right)\hfill \\ \hfill & ={\int }_0^t{W}_s\left(-\lambda e^{-\lambda (t-s)}{(t-s)}^{\frac{1}{2}}+\frac{1}{2}{e}^{-\lambda (t-s)}{(t-s)}^{-\frac{1}{2}}\right)ds\hfill \\ \hfill & =J_1\left(t\right)+J_2\left(t\right),\hfill \end{array}$$

where

$$J_1\left(t\right)=-\lambda {\int }_0^t{W}_s{e}^{-\lambda (t-s)}{(t-s)}^{\frac{1}{2}}ds,J_2\left(t\right)=\frac{1}{2}{\int }_0^t{W}_s{e}^{-\lambda (t-s)}{(t-s)}^{-\frac{1}{2}}ds.$$

Evidently, $J_1\left(t\right)$ is differentiable with probability 1, and

$$J_1^{\prime }\left(t\right)=-\lambda {\int }_0^t{W}_s\left(-\lambda e^{-\lambda (t-s)}{(t-s)}^{\frac{1}{2}}+\frac{1}{2}{e}^{-\lambda (t-s)}{(t-s)}^{-\frac{1}{2}}\right)ds.$$

However, if we try to differentiate $J_2\left(t\right)$ formally, we obtain

$$J_2^{\prime }\left(t\right)=\frac{1}{2}\frac{d}{dt}\left({\int }_0^t{W}_s{e}^{-\lambda (t-s)}{(t-s)}^{-\frac{1}{2}}ds\right),$$

but this derivative is non-existent for any $t>0$.

3.2. Asymptotic Behavior of Variance of TFBMII at Zero

Let us consider TFBMII.

Lemma 3.

(i)

For any $H\in (0,1)$

$$\mathbf{E}\left[B_{H,\lambda }^{II}{\left(t\right)}^2\right]\sim \frac{\mathsf{\Gamma }{\left(H+\frac{1}{2}\right)}^2}{2Hsin\left(\pi H\right)\mathsf{\Gamma }\left(2H\right)}{t}^{2H},\mathrm{as}t\downarrow 0.$$

(16)

(ii)

For $H=1$

$$\mathbf{E}\left[B_{H,\lambda }^{II}{\left(t\right)}^2\right]\sim \frac{t^2}{4}logt,\mathrm{as}t\downarrow 0.$$

(iii)

For any $H>1$

$$\mathbf{E}\left[B_{H,\lambda }^{II}{\left(t\right)}^2\right]\sim \frac{\left(2H-1\right)\mathsf{\Gamma }\left(2H\right)}{2^{2H+1}{\lambda }^{2H-2}(H-1)}{t}^2,\mathrm{as}t\downarrow 0.$$

(17)

The proof of Lemma 3 is given in Appendix A.

Remark 4.

The comparison of Lemmas 1 and 3 shows that the variances of TFBM and TFBMII demonstrate the same behavior at zero for $H\in (0,1]$. For $H>1$, they also behave similarly, as $C{t}^2$, but the constants are different, so the variance of TFBMII is bigger. Namely, the difference between constants is

$$\frac{\left(2H-1\right)\mathsf{\Gamma }\left(2H\right)}{2^{2H+1}{\lambda }^{2H-2}(H-1)}-\frac{\mathsf{\Gamma }\left(2H\right)}{2^{2H+1}{\lambda }^{2H-2}(H-1)}=\frac{\mathsf{\Gamma }\left(2H\right)}{2^{2H}{\lambda }^{2H-2}}.$$

(18)

In order to explain this fact, we note that, for the same Wiener process W, TFBM in (2) and TFBMII in (3) are related as

$$B_{H,\lambda }^{II}\left(t\right)=B_{H,\lambda }^I\left(t\right)+\lambda Y_{H,\lambda }\left(t\right),$$

(19)

where

$$\begin{array}{cc}\hfill Y_{H,\lambda }\left(t\right)& ={\int }_0^t{\int }_{\mathbb{R}}{(u-s)}_{+}^{H-\frac{1}{2}}{e}^{-\lambda {(u-s)}_{+}}d{W}_{s}du\hfill \\ \hfill & ={\int }_0^t{\int }_{\mathbb{R}}\left({(u-s)}_{+}^{H-\frac{1}{2}}{e}^{-\lambda {(u-s)}_{+}}-{(-s)}_{+}^{H-\frac{1}{2}}{e}^{-\lambda {(-s)}_{+}}\right)d{W}_{s}du\hfill \\ \hfill & +{\int }_0^t{\int }_{\mathbb{R}}{(-s)}_{+}^{H-\frac{1}{2}}{e}^{-\lambda {(-s)}_{+}}d{W}_{s}du={\int }_0^t{B}_{H,\lambda }^I\left(s\right)ds+t{\eta }_{H,\lambda },\hfill \end{array}$$

where

$${\eta }_{H,\lambda }={\int }_{-\infty }^0{(-s)}^{H-\frac{1}{2}}{e}^{-\lambda (-s)}d{W}_s.$$

(20)

Now it becomes clear that the difference (18) corresponds to the variance of the random variable $\lambda {\eta }_{H,\lambda }$, which equals

$${\lambda }^2\mathbf{E}\left[{\eta }_{H,\lambda }^2\right]={\lambda }^2{\int }_0^{\infty }{x}^{2H-1}{e}^{-2\lambda x}dx=\frac{\mathsf{\Gamma }\left(2H\right)}{2^{2H}{\lambda }^{2H-2}}.$$

The next result follows from Lemma 3 and the convergence (10). The proof is similar to that of Lemma 2, and therefore, it is omitted.

Lemma 4.

(i)

For any $H\in (0,\frac{1}{2})$ and for all $t,s\in {\mathbb{R}}_{+}$

$$\mathbf{E}\left[|B_{H,\lambda }^{II}\left(t\right)-B_{H,\lambda }^{II}\left(s\right){|}^2\right]\le C\left({\left|t-s\right|}^{2H}\vee \left|t-s\right|\right).$$

(ii)

For any $H\in [\frac{1}{2},1)$ and for all $t,s\in {\mathbb{R}}_{+}$

$$\mathbf{E}\left[|B_{H,\lambda }^{II}\left(t\right)-B_{H,\lambda }^{II}\left(s\right){|}^2\right]\le C\left({\left|t-s\right|}^{2H}\wedge \left|t-s\right|\right).$$

(iii)

If $H=1$, then for all $t,s\in {\mathbb{R}}_{+}$

$$\mathbf{E}\left[|B_{H,\lambda }^{II}\left(t\right)-B_{H,\lambda }^{II}\left(s\right){|}^2\right]\le C\left({\left|t-s\right|}^2\left|log\left|t-s\right|\right|\wedge \left|t-s\right|\right).$$

(iv)

For any $H>1$ and for all $t,s\in {\mathbb{R}}_{+}$

$$\mathbf{E}\left[|B_{H,\lambda }^{II}\left(t\right)-B_{H,\lambda }^{II}\left(s\right){|}^2\right]\le C\left({\left|t-s\right|}^2\wedge \left|t-s\right|\right).$$

Remark 5.

We see that the incremental variances of TFBMII demonstrate similar behavior to that of TFBM for small $\left|t-s\right|$, but for large $\left|t-s\right|$, these two processes behave differently: TFBM has bounded incremental variances, while the incremental variance of TFBMII behaves as $O\left(\right|t-s\left|\right)$ for all $H>0$.

Let us investigate the Hölder and Lipschitz continuity of TFBMII. These properties are similar to those of TFBM.

1. If $H\in (0,1]$, then $B_{H,\lambda }^{II}$ satisfies the Hölder condition of any order $\gamma \in (0,H)$ on any interval $[0,T]$. Moreover, it fails to satisfy this condition for $\gamma >H$. Indeed, if $H\in (0,1)$ for any $T>0$, we have the bound

$$\mathbf{E}\left[|B_{H,\lambda }^{II}\left(t\right)-B_{H,\lambda }^{II}\left(s\right){|}^2\right]\le C{\left|t-s\right|}^{2H-ϵ},$$

for any $H\in (0,1]$, $ϵ\in (0,H)$ and $t\in [0,T]$ (the constant C may depend on H, ϵ, and T). Therefore, the announced Hölder continuity follows from ([22], Theorem 1).

2. In order to investigate path regularity properties in more detail, we recall that, by representation (19),

$$B_{H,\lambda }^{II}\left(t\right)=B_{H,\lambda }^I\left(t\right)+\lambda Y_{H,\lambda }\left(t\right)=B_{H,\lambda }^I\left(t\right)+\lambda {\int }_0^t{B}_{H,\lambda }^I\left(s\right)ds+\lambda {\eta }_{H,\lambda }t,$$

where the random variable ${\eta }_{H,\lambda }$ is defined by (20). Obviously, for any $H>0$, the process $Y_{H,\lambda }$ is differentiable with

$$\frac{d}{dt}{Y}_{H,\lambda }\left(t\right)=B_{H,\lambda }^I\left(t\right)+{\eta }_{H,\lambda }.$$

Moreover, if $B_{H,\lambda }^I\left(t\right)$ is p times differentiable, $p\in \mathbb{N}$, then the process $Y_{H,\lambda }\left(t\right)$ is $(p+1)$ times differentiable and

$$\frac{d^{p+1}}{d{t}^{p+1}}{Y}_{H,\lambda }\left(t\right)=\frac{d^p}{d{t}^p}{B}_{H,\lambda }^I\left(t\right).$$

Thus, we can conclude that the process $B_{H,\lambda }^{II}\left(t\right)$ has similar continuity and differentiability properties to that of $B_{H,\lambda }^I\left(t\right)$, namely

• for $H=1$, the process $B_{1,\lambda }^{II}\left(t\right)$ is not continuously differentiable for any $t\ge 0$,
• for $H>1$, the process $B_{H,\lambda }^{II}\left(t\right)$ is continuously differentiable for any $t\ge 0$; moreover, it is $p\in \mathbb{N}$ times differeniable for $H>p$.

4. Asymptotic Growth of the Trajectories of TFBM and TFBMII with Probability 1

In this section, we study the asymptotic growth of $B_{H,\lambda }^I\left(t\right)$ and $B_{H,\lambda }^{II}\left(t\right)$ as $t\to \infty$.

4.1. Asymptotic Growth of the Trajectories of TFBM with Probability 1

Let $\left\{b_k,k\ge 0\right\}$ be a sequence such that $b_0=0$, $b_{k+1}-b_k\ge 1$, and let $a\left(t\right)>0$ be an increasing continuous function such that $a\left(t\right)\to \infty$ as $t\to \infty$, $a_k=a\left(b_k\right)$.

Note that according to Lemma 2 there exist $K_1$ and $K_2$ such that

$$\begin{array}{c}\underset{t\in [b_k,b_{k+1}]}{sup}{\left(\mathbf{E}|B_{H,\lambda }^I\left(t\right){|}^2\right)}^{1/2}\le K_1,\end{array}$$

(21)

$$\begin{array}{c}\underset{\begin{array}{c}t,s\in [b_k,b_{k+1}]\\ \left|t-s\right|\le h\end{array}}{sup}{\left(\mathbf{E}{|B_{H,\lambda }^I\left(t\right)-B_{H,\lambda }^I\left(s\right)|}^2\right)}^{1/2}\le K_2{l}_k{h}^{H\wedge 1},\end{array}$$

(22)

for any $h>0$, where

$$l_k=\left\{\begin{array}{cc}1,\hfill & H\ne 1,\hfill \\ {(1\vee log{b}_{k+1})}^{1/2},\hfill & H=1,\hfill \end{array}\right.k\ge 0.$$

Theorem 2.

Let $\beta =H\wedge 1$. Assume that there exists $0<\gamma \le 1$ such that

$$\rho :=\sum _{k=0}^{\infty }\frac{{\left(b_{k+1}-b_k\right)}^{\gamma }{l}_k^{\gamma /\beta }}{a_k}<\infty .$$

(23)

Then for all $t>0$, we have with probability 1 the following bound

$$\underset{s\in [0,t]}{sup}\left|B_{H,\lambda }^I\left(s\right)\right|\le a\left(t\right)\xi ,$$

where ξ is a non-negative random variable whose distribution has the tail admitting the following upper bound: for any $\theta \in (0,1)$ and any $u>\alpha$

$$\mathbf{P}(\xi >u)\le 2^{\frac{2}{\beta }-1}exp\left\{-\frac{u^2{(1-\theta )}^2}{2{\alpha }^2}\right\}{A}_1\left(\theta ,\gamma \right),$$

where

$$\alpha =K_1\sum _{k=0}^{\infty }{a}_k^{-1},A_1(\theta ,\gamma )=exp\left\{\frac{\rho }{\gamma \alpha }{K}_1^{1-\gamma /\beta }{K}_2^{\gamma /\beta }{\left(\frac{2^{2/\beta -1}}{{\theta }^{1/\beta }}\right)}^{\gamma }\right\}.$$

Moreover, for any $u>\alpha$

$$\mathbf{P}\left\{\xi >u\right\}\le 2^{\frac{2}{\beta }-1}\sqrt{e}exp\left\{-\frac{u^2}{2{\alpha }^2}\right\}{A}_1\left(1-\sqrt{1-\frac{{\alpha }^2}{u^2}},\gamma \right).$$

Proof. 

We will apply Theorem A1 from Appendix A. Let us verify its assumptions. According to (22), the assumption $\left(i\right)$ holds with $c_k=K_2{l}_k$. Taking into account (21) and (23), we see that $\left(ii\right)$ holds with $m_k=K_1$, since

$$\sum _{k=0}^{\infty }\frac{m_k}{a_k}=K_1\sum _{k=0}^{\infty }\frac{1}{a_k}\le K_1\sum _{k=0}^{\infty }\frac{{\left(b_{k+1}-b_k\right)}^{\gamma }{l}_k^{\gamma /\beta }}{a_k}=K_1\rho <\infty .$$

Finally, the assumption $\left(iii\right)$ of Theorem A1 is satisfied too, because

$$\begin{array}{cc}\hfill \sum _{k=0}^{\infty }\frac{m_k^{1-\gamma /\beta }{(b_{k+1}-b_k)}^{\gamma }{c}_k^{\gamma /\beta }}{a_k}& =K_1^{1-\gamma /\beta }{K}_2^{\gamma /\beta }\sum _{k=0}^{\infty }\frac{{(b_{k+1}-b_k)}^{\gamma }{l}_k^{\gamma /\beta }}{a_k}\hfill \\ \hfill & =K_1^{1-\gamma /\beta }{K}_2^{\gamma /\beta }\rho <\infty .\hfill \end{array}$$

by (23). Thus, the result follows from Theorem A1.  □

Now, we are ready to prove the first main result of this section.

Theorem 3.

For any $\delta >0$, there exists a non-negative random variable $\xi =\xi \left(\delta \right)$ such that for all $t>0$

$$\underset{s\in [0,t]}{sup}\left|B_{H,\lambda }^I\left(s\right)\right|\le \left(t^{\delta }\vee 1\right)\xi a.s.,$$

and there exist positive constants $C_1=C_1\left(\delta \right)$ and $C_2=C_2\left(\delta \right)$ such that for all $u>0$

$$\mathbf{P}(\xi >u)\le C_1{e}^{-C_2{u}^2}.$$

(24)

Let $\alpha =\frac{K_1}{1-e^{-\delta }}$. Then for any $\gamma >0$ there exists a constant $C_3>0$ such that for all $u>\alpha$

$$\mathbf{P}(\xi >u)\le 2^{\frac{2}{\beta }-1}\sqrt{e}exp\left\{-\frac{u^2}{2{\alpha }^2}\right\}exp\left\{C_3{\left(1-\sqrt{1-\frac{{\alpha }^2}{u^2}}\right)}^{-\gamma /\beta }\right\}.$$

Proof. 

Let us put in Theorem 2 $a\left(t\right)=t^{\delta }\vee 1$, $b_0=0$, $b_k=e^k$, $k\ge 1$, and arbitrary $\gamma \in (0,\delta )$ and $\theta \in (0,1)$. Then $a_k=e^{k\delta }$, $k\ge 0$, and

$$\sum _{k=0}^{\infty }\frac{{\left(b_{k+1}-b_k\right)}^{\gamma }{l}_k^{\gamma /\beta }}{a_k}\le \sum _{k=0}^{\infty }\frac{{\left(e^{k+1}-e^k\right)}^{\gamma }{(k+1)}^{\frac{\gamma }{2\beta }}}{a_k}\le e^{\gamma }\sum _{k=0}^{\infty }{e}^{k(\gamma -\delta )}{(k+1)}^{\frac{\gamma }{2\beta }}<\infty ,$$

hence, the assumption (23) of Theorem 2 is satisfied. Hence, the result follows from Theorem 2, if we choose

$$\begin{array}{c}{C}_1=2^{\frac{2}{\beta }-1}{A}_1\left(\theta ,\gamma \right),C_2=\frac{{(1-\theta )}^2}{2{\alpha }^2},\\ C_3=\frac{\rho }{\gamma \alpha }{K}_1^{1-\gamma /\beta }{K}_2^{\gamma /\beta }{2}^{2\gamma /\beta -\gamma },\end{array}$$

and note that

$$\alpha =K_1\sum _{k=0}^{\infty }{a}_k^{-1}=K_1\sum _{k=0}^{\infty }{e}^{-k\delta }=\frac{K_1}{1-e^{-\delta }}.$$

 □

Remark 6.

For FBM $B^H=\{B_t^H,t\ge 0\}$, the following upper bound was established in [19]

$$\underset{s\in [0,t]}{sup}\left|B_s^H\right|\le \left(1\vee \left(t^H{\left({log}^{+}t\right)}^p\right)\right)\xi \left(p\right)$$

(25)

for any $p>0$. We see that TBFM demonstrates much slower asymptotic growth.

4.2. Asymptotic Growth of the Trajectories of TFBMII with Probability 1

Now, let us consider TFBMII and establish its a.s. asymptotic growth using similar ideas as above. Assume as before that $\left\{b_k,k\ge 0\right\}$ is a sequence such that $b_0=0$, $b_{k+1}-b_k\ge 1$, and $a\left(t\right)>0$ is an increasing continuous function such that $a\left(t\right)\to \infty$ as $t\to \infty$, $a_k=a\left(b_k\right)$. In order to apply Theorem A1, we note that

$$\underset{t\in [b_k,b_{k+1}]}{sup}{\left(\mathbf{E}{\left|B_{H,\lambda }^{II}\left(t\right)\right|}^2\right)}^{1/2}\le L_1{b}_{k+1}^{1/2}.$$

(26)

by Lemma 4. Moreover, the following upper bounds hold.

Lemma 5.

Let $h>0$.

(i)

If $H\in (0,\frac{1}{2})$, then there exists $L_2>0$ such that

$$\underset{\begin{array}{c}t,s\in [b_k,b_{k+1}]\\ \left|t-s\right|\le h\end{array}}{sup}{\left(\mathbf{E}{|B_{H,\lambda }^{II}\left(t\right)-B_{H,\lambda }^{II}\left(s\right)|}^2\right)}^{1/2}\le L_2{(b_{k+1}-b_k)}^{\frac{1}{2}-H}{h}^H,k\ge 0.$$

(27)

(ii)

If $H\ge \frac{1}{2}$, then there exists $L_2>0$ such that

$$\underset{\begin{array}{c}t,s\in [b_k,b_{k+1}]\\ \left|t-s\right|\le h\end{array}}{sup}{\left(\mathbf{E}{|B_{H,\lambda }^{II}\left(t\right)-B_{H,\lambda }^{II}\left(s\right)|}^2\right)}^{1/2}\le L_2{h}^{\frac{1}{2}},k\ge 0.$$

(28)

Proof. 

We fix some $k\ge 0$ and $t,s\in [b_k,b_{k+1}]$. Recall that $b_{k+1}-b_k\ge 1$.

$\left(i\right)$ Let $H\in (0,\frac{1}{2})$. By Lemma 4 $\left(i\right)$, we have:

(1)

If $\left|t-s\right|\le 1$, then

$$\mathbf{E}|B_{H,\lambda }^{II}\left(t\right)-B_{H,\lambda }^{II}\left(s\right){|}^2\le {C\left|t-s\right|}^{2H}\le C{\left|t-s\right|}^{2H}{(b_{k+1}-b_k)}^{1-2H}.$$

(2)

If $\left|t-s\right|>1$, then

$$\begin{array}{cc}\hfill \mathbf{E}|B_{H,\lambda }^{II}\left(t\right)-B_{H,\lambda }^{II}\left(s\right){|}^2& \le C\left|t-s\right|={C\left|t-s\right|}^{2H}{\left|t-s\right|}^{1-2H}\hfill \\ \hfill & \le {C\left|t-s\right|}^{2H}{(b_{k+1}-b_k)}^{1-2H}.\hfill \end{array}$$

Thus, in both cases, we arrive at (27).

$\left(ii\right)$ From Lemma 4 $\left(ii\right)$–$\left(iv\right)$, we see that for any $H\ge 1/2$

$$\mathbf{E}|B_{H,\lambda }^{II}\left(t\right)-B_{H,\lambda }^{II}\left(s\right){|}^2\le C\left|t-s\right|,$$

whence (28) follows.  □

Further, let us formulate an auxiliary theorem.

Theorem 4.

Let $\nu =H\wedge \frac{1}{2}$. Assume that

$$\sum _{k=0}^{\infty }\frac{b_{k+1}^{1/2}}{a_k}<\infty .$$

(29)

Then for all $t>0$ we have with probability 1 the following bound,

$$\underset{s\in [0,t]}{sup}\left|B_{H,\lambda }^{II}\left(s\right)\right|\le a\left(t\right)\xi ,$$

where ξ is a non-negative random variable whose distribution has the tail admitting the following upper bound: for any $\theta ,\gamma \in (0,1)$ and any $u>\alpha$

$$\mathbf{P}(\xi >u)\le 2^{\frac{2}{\nu }-1}exp\left\{-\frac{u^2{(1-\theta )}^2}{2{\alpha }^2}\right\}{A}_2\left(\theta ,\gamma \right),$$

where $\alpha =L_1{\sum }_{k=0}^{\infty }\frac{b_{k+1}^{1/2}}{a_k}$,

$$A_2(\theta ,\gamma )=exp\left\{\frac{1}{\gamma \alpha }{L}_1^{1-\gamma /\nu }{L}_2^{\gamma /\nu }\left(\sum _{k=0}^{\infty }\frac{b_{k+1}^{\frac{1}{2}-\frac{\gamma }{2\nu }}{(b_{k+1}-b_k)}^{\frac{\gamma }{2\nu }}}{a_k}\right){\left(\frac{2^{2/\nu -1}}{{\theta }^{1/\nu }}\right)}^{\gamma }\right\}.$$

Moreover, for any $u>\alpha$

$$\mathbf{P}\left\{\xi >u\right\}\le 2^{\frac{2}{\nu }-1}\sqrt{e}exp\left\{-\frac{u^2}{2{\alpha }^2}\right\}{A}_2\left(1-\sqrt{1-\frac{{\alpha }^2}{u^2}},\gamma \right).$$

Proof. 

Let us verify the assumptions $\left(i\right)$–$\left(iii\right)$ of Theorem A1.

$\left(i\right)$ According to Lemma 5, $\left(i\right)$ holds with $\beta =\nu$, $c_k=L_2{(b_{k+1}-b_k)}^{\frac{1}{2}-\nu }$.

$\left(ii\right)$ In our case $m_k=L_1{b}_{k+1}^{1/2}$, by (26). Therefore $\left(ii\right)$ follows from (29):

$$\sum _{k=0}^{\infty }\frac{m_k}{a_k}=L_1\sum _{k=0}^{\infty }\frac{b_{k+1}^{1/2}}{a_k}<\infty .$$

$\left(iii\right)$ By (29), we obtain

$$\begin{array}{cc}\hfill \sum _{k=0}^{\infty }\frac{m_k^{1-\gamma /\nu }{(b_{k+1}-b_k)}^{\gamma }{c}_k^{\gamma /\nu }}{a_k}& =L_1^{1-\gamma /\nu }{L}_2^{\gamma /\nu }\sum _{k=0}^{\infty }\frac{b_{k+1}^{\frac{1}{2}-\frac{\gamma }{2\nu }}{(b_{k+1}-b_k)}^{\frac{\gamma }{2\nu }}}{a_k}\hfill \\ \hfill & \le L_1^{1-\gamma /\nu }{L}_2^{\gamma /\nu }\sum _{k=0}^{\infty }\frac{b_{k+1}^{\frac{1}{2}}}{a_k}<\infty .\hfill \end{array}$$

Now, the result follows from Theorem A1.  □

The following theorem produces the second main result of the present section, namely the asymptotic growth with probability 1 for TFBMII.

Theorem 5.

For any $p>1$, there exists a non-negative random variable $\xi =\xi \left(p\right)$ such that for all $t>0$

$$\underset{s\in [0,t]}{sup}\left|B_{H,\lambda }^{II}\left(s\right)\right|\le \left(\left(t^{\frac{1}{2}}{\left({log}^{+}t\right)}^p\right)\vee 1\right)\xi a.s.,$$

(30)

and there exist positive constants $C_1=C_1\left(p\right)$ and $C_2=C_2\left(p\right)$ such that for all $u>0$

$$\mathbf{P}(\xi >u)\le C_1{e}^{-C_2{u}^2}.$$

Let $\alpha =L_1{e}^{1/2}{\sum }_{k=0}^{\infty }{k}^{-p}$, $\nu =H\wedge \frac{1}{2}$. Then for any $\gamma >0$, there exists a constant $C_3>0$ such that for all $u>\alpha$

$$\mathbf{P}(\xi >u)\le 2^{\frac{2}{\nu }-1}\sqrt{e}exp\left\{-\frac{u^2}{2{\alpha }^2}\right\}exp\left\{C_3{\left(1-\sqrt{1-\frac{{\alpha }^2}{u^2}}\right)}^{-\gamma /\nu }\right\}.$$

Proof. 

The result follows from Theorem 4 if we put $a\left(t\right)=\left(t^{\frac{1}{2}}{\left({log}^{+}t\right)}^p\right)\vee 1$, $b_0=0$, $b_k=e^k$, $k\ge 1$, and arbitrary $\gamma ,\theta \in (0,1)$. Indeed, then $a_0=1$, $a_k=e^{k/2}{k}^p$, $k\ge 1$, and the condition (29) holds:

$$\sum _{k=0}^{\infty }\frac{b_{k+1}^{1/2}}{a_k}=\sum _{k=0}^{\infty }\frac{e^{\frac{1}{2}(k+1)}}{e^{\frac{k}{2}}{k}^p}=e^{\frac{1}{2}}\sum _{k=0}^{\infty }{k}^{-p}<\infty .$$

□

Remark 7.

The result of Theorem 5 cannot be improved substantially in a sense that one cannot obtain the power of t less than $\frac{1}{2}$ in the right-hand side of (30). Indeed, the distribution of the random variable $B_{H,\lambda }^{II}\left(t\right)$ is Gaussian with a zero mean and with variance ${\left(C_t^{II}\right)}^2{t}^{2H}$, see the statement (2) of Proposition 1. Then $B_{H,\lambda }^{II}\left(t\right)=C_t^{II}{t}^H\xi$, where ξ is a standard normal random variable. But $C_t^{II}{t}^H\sim C{t}^{\frac{1}{2}}$ as $t\to \infty$, see (10).

Remark 8.

Note that TFBMII demonstrates faster asymptotic growth at infinity than TFBM. Let us compare the result of Theorem 5 with the corresponding upper bound for FBM (25). We see that for $H>1/2$, the asymptotic growth of TFBMII is slower compared to that of FBM. However, for $H<1/2$, it is faster.

5. Drift Parameter Estimation in the Vasicek-Type Model

5.1. Strongly Consistent Drift Estimation in the Fractional Tempered Vasicek Model

Our focus in this section is on the statistical inference of the tempered fractional Vasicek model, which is described by the following stochastic differential equation:

$$Y_t=y_0+{\int }_0^t(a+b{Y}_s)ds+\sigma X_t,t>0,Y_0=y_0\in \mathbb{R},$$

(31)

where $a\in \mathbb{R}$, $b>0$, $\sigma >0$, $X=X_{H,\lambda }$ is TFBM $B_{H,\lambda }^I$ or TFBMII $B_{H,\lambda }^{II}$. Equivalently, $Y=\{Y_t,t\ge 0\}$ is the process given explicitly by

$$Y_t=y_0{e}^{bt}-\frac{a}{b}\left(1-e^{bt}\right)+\sigma {\int }_0^t{e}^{b(t-s)}d{X}_s,$$

(32)

where the stochastic integral with respect to TFBMs is defined in [24].

Let us consider the estimation of unknown drift parameter $\theta =(a,b)\in \mathbb{R}\times (0,\infty )$ in the Vasicek model (31) driven by TFBM or TFBMII.

We define the estimator ${\widehat{\theta }}_T=({\widehat{a}}_T,{\widehat{b}}_T)$ as follows

$$\begin{array}{c}{\widehat{a}}_T=\frac{(Y_T-y_0)\left({\int }_0^T{Y}_t^{2}dt-\frac{1}{2}(Y_T+y_0){\int }_0^T{Y}_{t}dt\right)}{T{\int }_0^T{Y}_t^{2}dt-{\left({\int }_0^T{Y}_{t}dt\right)}^2},\end{array}$$

(33)

$$\begin{array}{c}{\widehat{b}}_T=\frac{(Y_T-y_0)\left(\frac{1}{2}T(Y_T+y_0)-{\int }_0^T{Y}_{t}dt\right)}{T{\int }_0^T{Y}_t^{2}dt-{\left({\int }_0^T{Y}_{t}dt\right)}^2}.\end{array}$$

(34)

It is worth noting that both ${\widehat{a}}_T$ and ${\widehat{b}}_T$ do not depend on the diffusion parameters $H>0$, $\lambda >0$, and $\sigma >0$, which may be unknown.

Remark 9.

The estimator $\widehat{\theta }$ given by (33)–(34) can be thought of as the least squares estimator (LSE). Let us present its (formal) derivation. If we assume that the derivative $\dot{Y}$ and the integral ${\int }_0^T{Y}_{t}d{Y}_t$ exist in some sense, then we can define LSE as the value of $(a,b)$, which minimizes the following functional

$$\begin{array}{cc}\hfill F(a,b)& ={\int }_0^T{\left({\dot{Y}}_t-a-b{Y}_t\right)}^{2}dt\hfill \\ \hfill & ={\int }_0^T{\dot{Y}}_t^{2}dt+a^{2}T+b^2{\int }_0^T{Y}_t^{2}dt-2a{\int }_0^{T}d{Y}_t-2b{\int }_0^T{Y}_{t}d{Y}_t+2ab{\int }_0^T{Y}_{t}dt.\hfill \end{array}$$

It is not hard to see that the minimum is achieved at the following point:

$$\begin{array}{c}{\widehat{a}}_T=\frac{{\int }_0^{T}d{Y}_t{\int }_0^T{Y}_t^{2}dt-{\int }_0^T{Y}_{t}dt{\int }_0^T{Y}_{t}d{Y}_t}{T{\int }_0^T{Y}_t^{2}dt-{\left({\int }_0^T{Y}_{t}dt\right)}^2},\end{array}$$

(35)

$$\begin{array}{c}{\widehat{b}}_T=\frac{T{\int }_0^T{Y}_{t}d{Y}_t-{\int }_0^{T}d{Y}_t{\int }_0^T{Y}_{t}dt}{T{\int }_0^T{Y}_t^{2}dt-{\left({\int }_0^T{Y}_{t}dt\right)}^2}.\end{array}$$

(36)

Note that these expressions do not contain the derivative $\dot{Y}$, which may not exist. Moreover, if the process Y is γ-Hölder continuous on $[0,T]$ with any $\gamma >\frac{1}{2}$, then the integral ${\int }_0^T{Y}_{t}d{Y}_t$ can be defined in Young’s sense. In this case, it admits the representation

$${\int }_0^T{Y}_{t}d{Y}_t=\frac{1}{2}\left(Y_T^2-Y_0^2\right).$$

Substitution of these representations into the Formulas (35)and (36) leads to the estimators (33) and (34).

Certainly, we cannot guarantee the required Hölder continuity of order greater than $\frac{1}{2}$ in the case $H<\frac{1}{2}$ (see Remarks 3 and 5). Nevertheless, it turns out that the estimators defined by (33) and (34) remain strongly consistent for all $H>0$.

The following theorem is the main result of this section.

Theorem 6.

The estimator $({\widehat{a}}_T,{\widehat{b}}_T)$ is strongly consistent, i.e.,

$${\widehat{a}}_T\to a,{\widehat{b}}_T\to ba.s.whenT\to \infty .$$

In order to prove the strong consistency of the estimator $({\widehat{a}}_T,{\widehat{b}}_T)$, we need to study the asymptotic behavior of all processes and integrals involved into the right-hand sides of (33) and (34).

Our reasoning will be based substantially on the results of Section 4. Let

$$a\left(t\right)=\left\{\begin{array}{cc}{t}^{\delta }\vee 1,\hfill & \mathrm{for}X=B_{H,\lambda }^I,\hfill \\ \left(t^{\frac{1}{2}}{\left({log}^{+}t\right)}^p\right)\vee 1,\hfill & \mathrm{for}X=B_{H,\lambda }^{II},\hfill \end{array}\right.$$

where $\delta \in (0,1)$ and $p>1$ are arbitrarily chosen constants. Then, according to Theorems 3 and 5, we have the following upper bound: for all $t>0$

$$\underset{0\le s\le t}{sup}\left|X_s\right|\le a\left(t\right)\xi \mathrm{a}.\mathrm{s}.,$$

(37)

where $\xi$ is a non-negative random variable satisfying (24).

Therefore,

$${\int }_0^{\infty }{e}^{-bs}\left|X_s\right|ds<\infty \mathrm{a}.\mathrm{s}.,$$

and the following random variables are well-defined:

$$Z_{\infty }:={\int }_0^{\infty }{e}^{-bs}{X}_{s}ds,\zeta :=y_0+\frac{a}{b}+\sigma b{Z}_{\infty }.$$

Lemma 6.

The following convergences hold a.s. when $T\to \infty$:

$$\begin{array}{c}{e}^{-bT}{Y}_T\to \zeta ,\end{array}$$

(38)

$$\begin{array}{c}{e}^{-bT}{\int }_0^T{Y}_{t}dt\to \frac{1}{b}\zeta ,\end{array}$$

(39)

$$\begin{array}{c}{e}^{-bT}{\int }_0^T\left|Y_t\right|dt\to \frac{1}{b}\left|\zeta \right|,\end{array}$$

(40)

$$\begin{array}{c}{e}^{-2bT}{\int }_0^T{Y}_t^{2}dt\to \frac{1}{2b}{\zeta }^2,\end{array}$$

(41)

$$\begin{array}{c}{T}^{-1}{e}^{-bT}{\int }_0^T{Y}_{t}tdt\to \frac{1}{b}\zeta .\end{array}$$

(42)

Proof. 

Recall that according to (32), the process Y can be represented as follows:

$$Y_t=y_0{e}^{bt}-\frac{a}{b}\left(1-e^{bt}\right)+\sigma e^{bt}{U}_t,$$

(43)

where

$$U_t={\int }_0^t{e}^{-bs}d{X}_s.$$

Using the integration-by-parts formula, we can write

$$U_T=e^{-bT}{X}_T+b{\int }_0^T{e}^{-bs}{X}_{s}ds=e^{-bT}{X}_t+b{Z}_T,$$

(44)

where

$$Z_T={\int }_0^T{e}^{-bs}{X}_{s}ds.$$

It follows from the upper bound (37) that

$$e^{-bT}{X}_T\to 0\mathrm{a}.\mathrm{s}.\mathrm{when}T\to \infty .$$

(45)

Furthermore, by the dominated convergence theorem, we see that

$$Z_T\to Z_{\infty }\mathrm{a}.\mathrm{s}.\mathrm{when}T\to \infty ,$$

(46)

and also note that all these random variables are Gaussian. Consequently, (44)–(46) imply that

$$U_T\to b{Z}_{\infty }\mathrm{a}.\mathrm{s}.\mathrm{when}T\to \infty .$$

Taking into account the representation (43), we obtain the following passage to the limit:

$$e^{-bT}{Y}_T\to y_0+\frac{a}{b}+\sigma b{Z}_{\infty }=\zeta \mathrm{a}.\mathrm{s}.\mathrm{when}T\to \infty .$$

Thus, (38) is proved. Moreover, using l’Hôpital’s rule, we immediately obtain the convergences (39)–(42) from (38).

□

Lemma 7.

The following convergence holds:

$$T^{-1}{e}^{-bT}\left({\int }_0^T{Y}_t^{2}dt-\frac{1}{2}(Y_T+y_0){\int }_0^T{Y}_{t}dt\right)\to \frac{a}{2b}\zeta a.s.whenT\to \infty .$$

(47)

Proof. 

Indeed, using the stochastic differential Equation (31), we can write

$$\begin{array}{c}{\int }_0^T{Y}_t^{2}dt-\frac{1}{2}(Y_T+y_0){\int }_0^T{Y}_{t}dt\hfill \\ \hspace{1em}\hspace{1em}={\int }_0^T{Y}_t\left(y_0+at+b{\int }_0^t{Y}_{s}ds+\sigma X_t\right)dt\hfill \\ \hspace{1em}\hspace{1em}-\frac{1}{2}\left(2y_0+aT+b{\int }_0^T{Y}_{t}dt+\sigma X_T\right){\int }_0^T{Y}_{t}dt\hfill \\ \hspace{1em}\hspace{1em}=a\left({\int }_0^T{Y}_{t}tdt-\frac{1}{2}T{\int }_0^T{Y}_{t}dt\right)+b\left({\int }_0^T{Y}_t{\int }_0^t{Y}_{s}dsdt-\frac{1}{2}{\left({\int }_0^T{Y}_{t}dt\right)}^2\right)\hfill \\ \hspace{1em}\hspace{1em}+\sigma \left({\int }_0^T{Y}_t{X}_{t}dt-\frac{1}{2}{X}_T{\int }_0^T{Y}_{t}dt\right)\hfill \\ \hspace{1em}\hspace{1em}=:a{I}_a\left(T\right)+b{I}_b\left(T\right)+\sigma I_{\sigma }\left(T\right).\hfill \end{array}$$

It is not hard to see that ${\int }_0^T{Y}_t{\int }_0^t{Y}_{s}dsdt=\frac{1}{2}{\left({\int }_0^T{Y}_{t}dt\right)}^2$, so $I_b\left(T\right)\equiv 0$.

Furthermore, using the convergences (39) and (42), we obtain that

$$\begin{array}{cc}{T}^{-1}{e}^{-bT}{I}_a\left(T\right)& =T^{-1}{e}^{-bT}{\int }_0^T{Y}_{t}tdt-\frac{1}{2}{e}^{-bT}{\int }_0^T{Y}_{t}dt\hfill \\ \hfill & \to \frac{1}{b}\zeta -\frac{1}{2b}\zeta =\frac{1}{2b}\zeta \mathrm{a}.\mathrm{s}.\mathrm{when}T\to \infty .\hfill \end{array}$$

Now it remains to prove that $T^{-1}{e}^{-bT}{I}_{\sigma }\left(T\right)$ tends to zero a.s. when $T\to \infty$. Note that according to (37),

$$\frac{1}{T}\underset{0\le t\le T}{sup}\left|X_t\right|\to 0\mathrm{a}.\mathrm{s}.\mathrm{when}T\to \infty .$$

(48)

Therefore,

$$\begin{array}{cc}\hfill T^{-1}{e}^{-bT}\left|I_{\sigma }\left(T\right)\right|& \le T^{-1}{e}^{-bT}{\int }_0^T\left|Y_t{X}_t\right|dt+\frac{1}{2}{T}^{-1}{e}^{-bT}\left|X_T\right|{\int }_0^T\left|Y_t\right|dt\hfill \\ \hfill & \le \frac{3}{2}·\frac{1}{T}\underset{0\le t\le T}{sup}\left|X_t\right|e^{-bT}{\int }_0^T\left|Y_t\right|dt\to 0\mathrm{a}.\mathrm{s}.\mathrm{when}T\to \infty ,\hfill \end{array}$$

by (48) and (40).  □

Proof of Theorem 6. 

In order to prove the strong consistency of ${\widehat{b}}_T$, we rewrite it in the following form:

$${\widehat{b}}_T=\frac{\frac{1}{2}{e}^{-2bT}(Y_T^2-y_0^2)-e^{-bT}(Y_T-y_0)T^{-1}{e}^{-bT}{\int }_0^T{Y}_{t}dt}{e^{-2bT}{\int }_0^T{Y}_t^{2}dt-{\left(T^{-1/2}{e}^{-bT}{\int }_0^T{Y}_{t}dt\right)}^2}=:\frac{B\left(T\right)}{D\left(T\right)}.$$

(49)

It follows from (38) and (39) that only the term $\frac{1}{2}{e}^{-2bT}{Y}_T^2$ in the numerator converges a.s. to the non-zero limit. Namely,

$$B\left(T\right)\to \frac{1}{2}{\zeta }^2\mathrm{a}.\mathrm{s}.\mathrm{when}T\to \infty .$$

Further, we obtain from (39) and (41):

$$D\left(T\right)\to \frac{1}{2b}{\zeta }^2\mathrm{a}.\mathrm{s}.\mathrm{when}T\to \infty .$$

(50)

Thus, ${\widehat{b}}_T\to b$ a.s. when $T\to \infty$.

Now, let us consider ${\widehat{a}}_T$.

$${\widehat{a}}_T=\frac{e^{-bT}(Y_T-y_0)T^{-1}{e}^{-bT}\left({\int }_0^T{Y}_t^{2}dt-\frac{1}{2}(Y_T+y_0){\int }_0^T{Y}_{t}dt\right)}{e^{-2bT}{\int }_0^T{Y}_t^{2}dt-{\left(T^{-1/2}{e}^{-bT}{\int }_0^T{Y}_{t}dt\right)}^2}=:\frac{A\left(T\right)}{D\left(T\right)}.$$

(51)

It follows from (38) and (47) that

$$A\left(T\right)\to \frac{a}{2b}{\zeta }^2\mathrm{a}.\mathrm{s}.\mathrm{when}T\to \infty .$$

Combining this convergence with (51) and (50), we obtain that ${\widehat{a}}_T\to a$ a.s. when $T\to \infty$.  □

Remark 10.

In the paper [15], the strong consistency of the least-squares estimators of $(a,b)$ in the Vasicek model (31) driven by a centered Gaussian process X was proved under the following condition: there exist constants $c>0$ and $\gamma \in (0,1)$ such that for every $s,t\ge 0$,

$$X_0=0,\mathbf{E}\left[{(X_t-X_s)}^2\right]\le c{\left|t-s\right|}^{2\gamma }.$$

(52)

Using this result, one can derive strong consistency of $({\widehat{a}}_T,{\widehat{b}}_T)$ from Lemma 2 in the case $X=B_{H,\lambda }^I$ (for any positive H and λ) and from Lemma 4 in the case $X=B_{H,\lambda }^{II}$ for $H\ge \frac{1}{2}$, $\lambda >0$. Our approach, based on the asymptotic growth, allows us to prove the strong consistency also for the case $X=B_{H,\lambda }^{II}$, $H\in (0,\frac{1}{2})$, where we have different exponents γ in the condition (52) for small and for large values of $\left|t-s\right|$. Moreover, this approach can be extended to the Vasicek model with other driving processes (e.g., multifractional Brownian motion) that do not satisfy condition (52) but exhibit the asymptotic growth similar to TFBM; see the next subsection for details.

5.2. Possible Generalization and Related Models

The analysis of the above proofs shows that Theorem 6 can be extended to the case of the Vasicek model (31) with a more general class of driving noises X. Namely, $X=\{X_t,t\ge 0\}$ can be any continuous centered Gaussian process with $X_0=0$, for with the condition (37), it holds with some positive non-decreasing function a satisfying the assumptions:

$$\begin{array}{c}\frac{a\left(t\right)}{t}\to 0\mathrm{as}t\to \infty ,\end{array}$$

(53)

$$\begin{array}{c}{\int }_0^{\infty }a\left(t\right)e^{-bt}dt<\infty \mathrm{for}\mathrm{any}b>0.\end{array}$$

(54)

Remark 11.

The consistency of ${\widehat{b}}_T$ can be proved without assumption (53), because condition (54) is sufficient for this. Indeed, (54) together with monotonicity of a implies that

$$a\left(t\right)e^{-bt}=b{\int }_t^{\infty }a\left(t\right)e^{-bs}ds\le b{\int }_t^{\infty }a\left(s\right)e^{-bs}ds\to 0\mathrm{as}t\to \infty ,$$

which allows us to prove all convergences in Lemma 6 and then derive consistency of ${\widehat{b}}_T$ from this lemma. Establishing the consistency of the estimate ${\widehat{a}}_T$ where the convergence of (48) is used requires a more restrictive assumption.

Let us apply these remarks to two examples of noise, namely multifractional Brownian motion and the Gaussian Volterra process.

Example 1

(Multifractional Vasicek model). Let $H:{\mathbb{R}}_{+}\to [h_1,h_2]\subset (0,1)$ be a κ-Hölder continuous function. We consider the model (31) driven by the (harmonizable) multifractional Brownian motion with functional parameter H, which is defined by (see [25])

$$X\left(t\right)={\int }_{\mathbb{R}}\frac{e^{itu}-1}{{\left|u\right|}^{H\left(t\right)+1/2}}d\tilde{W}\left(u\right),t\ge 0,$$

where $\tilde{W}\left(du\right)$ is the “Fourier transform” of the white noise $W\left(du\right)$ that is a unique complex-valued random measure such that for all $f\in L^2\left(\mathbb{R}\right)$

$${\int }_{\mathbb{R}}f\left(u\right)W\left(du\right)={\int }_{\mathbb{R}}\widehat{f}\left(u\right)\tilde{W}\left(du\right)a.s.$$

According to ([18], Theorem 3.8), the multifractional Brownian motion satisfies the condition (37) with $a\left(t\right)=t^{h^{*}+\delta }\vee 1$, where $h^{*}={lim\; sup}_{t\to \infty }H\left(t\right)$ and $\delta >0$ is arbitrary. Thus, the assumptions (53) and (54) hold; this implies the strong consistency of the estimator $({\widehat{a}}_T,{\widehat{b}}_T)$ in this model.

Remark 12.

The estimation of the parameter b for known $a=0$ in the multifractional Vasicek model was studied in ([18], Section 4.2), where the strongly consistent estimator was constructed under additional assumption $min\left\{h_1,\kappa \right\}>\frac{1}{2}$.

Example 2

(Vasicek model driven by a Gaussian Volterra process with power-type kernel). Let $\alpha >-\frac{1}{2}$, $\beta \in \mathbb{R}$, $\gamma >-1$ be such numbers that $\alpha +\beta +\gamma >-\frac{3}{2}$. Let us consider the following Gaussian process (see [26] and the references cited therein)

$$X_t={\int }_0^t{s}^{\alpha }{\int }_s^t{u}^{\beta }{(u-s)}^{\gamma }dud{W}_s,t\ge 0.$$

(55)

According to ([26], Theorem 2.3), this process satisfies the growth condition (37) with

$$a\left(t\right)=\left(t^{\alpha +\beta +\gamma +\frac{3}{2}}{(log(1\vee logt))}^{1/2}\right)\vee 1.$$

Therefore, under an additional condition

$$\alpha +\beta +\gamma <-\frac{1}{2},$$

(56)

the assumptions (53) and (54) hold; hence, the Formulas (33)–(34) provide strongly consistent estimators for the parameters a and b of the Vasicek model (31) driven by the Gaussian Volterra process (55).

Remark 13.

The estimation of b for known $a=0$ was investigated in [26], Section 4. Note that, in this case, the additional condition (56) is not required, cf. Remark 11.

6. Conclusions

We have investigated the behavior of TFBM and TFBMII at zero and infinity. Asymptotic bounds with probability 1 for the rate of growth of trajectories of these processes have been established. As an application of these results, we have demonstrated the strong consistency of the least-square estimators for the unknown drift parameters within the non-ergodic Vasicek model driven by either TFBM or TFBMII. These statistical findings have also been extended to Vasicek-type models with other Gaussian processes, specifically with multifractional Brownian motion and the Gaussian Volterra process with a power-type kernel.

Our research opens up possibilities for future extensions in several directions. In future studies, we plan to investigate parameter estimation in the “ergodic” Vasicek model (where $b<0$). It is worth noting that even in the case of an FBM-driven Vasicek model, the least-squares approach does not work for $b<0$, necessitating the development of alternative methodologies for parameter estimation. Moreover, potential avenues for further investigation include exploring the asymptotic distributions of the estimators in both cases, $b>0$ and $b<0$. Another challenging open problem is the simultaneous estimation of three noise parameters, $\sigma$, H and $\lambda$, along with the drift parameters a and b.