1 Introduction
Consider the stochastic process of the form
where W is a Wiener process, $C(\alpha ,\beta ,\gamma )$ is a constant.
(1)
\[ {X_{t}}=C(\alpha ,\beta ,\gamma ){\int _{0}^{t}}{s^{\alpha }}{\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\hspace{0.1667em}d{W_{s}},\]Our assumptions on the values of powers ensuring the existence and smoothness of X are
The process X from (1) is a representative of the processes of the form
which are studied in [4]. Here $a(s)$, $b(s)$ and $c(s)$ are measurable functions $[0,T]\to [-\infty ,\infty ]$. Initially, in this paper we intended to apply the results of [4] to power functions. However, the results in [4] are directly applicable only if, in addition to (2), ${\alpha ^{-}}+{\beta ^{-}}+{\gamma ^{-}}<\frac{3}{2}$. (Here we use notation ${x^{-}}=\max (-x,0)$ and ${x^{+}}=\max (x,0)$. The condition above can be rewritten as $(0\wedge \alpha )+(0\wedge \beta )+(0\wedge \gamma )>-\frac{3}{2}$.) It turned out that for the power kernel we can formulate more specific and weaker conditions of smoothness and other properties of X that are finer than in the general case.
(3)
\[ {X_{t}}={\int _{0}^{t}}a(s){\int _{s}^{t}}b(u)\hspace{0.1667em}c(u-s)\hspace{0.1667em}du\hspace{0.1667em}d{W_{s}},\]Note that process (3) belongs to the class of processes with Volterra kernels, i.e., the processes of the form
Such processes are discussed in [1, 2]. They are the particular case of the processes with Fredholm kernels, which are studied in [1, 8].
As it is well known, a fractional Brownian motion ${B^{H}}$ with Hurst index $H\in \big(\frac{1}{2},1\big)$ admits the Molchan representation (see [5, Theorem 1.8.3] or [7, Theorem 5.2]):
\[ {B_{t}^{H}}=\big(H-\frac{1}{2}\big)\hspace{0.1667em}{c_{H}}\hspace{0.1667em}{\int _{0}^{t}}{s^{\frac{1}{2}-H}}{\int _{s}^{t}}{u^{H-\frac{1}{2}}}{(u-s)^{H-\frac{3}{2}}}\hspace{0.1667em}du\hspace{0.1667em}d{W_{s}},\]
where
\[ {c_{H}}={\left(\frac{2H\hspace{0.1667em}\Gamma (1.5-H)}{\Gamma (H+0.5)\hspace{0.1667em}\Gamma (2-2H)}\right)^{\hspace{-0.1667em}1/2}}.\]
Thus, a fractional Brownian motion is an example of the process of the form (1).Concerning the related results in this direction, Azmoodeh et al. provide [2] necessary and sufficient conditions for the Hölder continuity of Gaussian processes and, as an application, for Fredholm processes. They also provide necessary and sufficient conditions as well as sufficient-only conditions for Volterra processes and for self-similar Gaussian processes. However, the sufficient-only conditions for self-similar Gaussian process, which are stated in [2, Proposition 3], are not satisfied for the process (1), at least, for some values of parameters α, β and γ that satisfy (2). The Fredholm representations of Gaussian processes were also considered in [8].
Even though we consider the process X on the interval $[0,T]$, it can be defined by (1) on the infinite interval $[0,\infty )$. Compare X with other Gaussian process such as fractional Brownian motion ${B^{H}}$, sub-fractional Brownian motion $\{({B_{t}^{H}}-{B_{-t}^{H}})/\sqrt{2},\hspace{0.2778em}t\ge 0\}$, bifractional Brownian motion ${B^{H,K}}$, and mixed fractional Brownian motion ${B^{{H_{1}}}}+{B^{{H_{2}}}}$, $0<{H_{1}}<{H_{2}}<1$ (the processes of this kind are studied in [6]). Here ${B^{H}}$ is a fractional Brownian motion on $\mathbb{R}$, ${B^{{H_{1}}}}$ and ${B^{{H_{2}}}}$ are two independent fractional Brownian motions with different Hurst indices. All these processes except ${B^{{H_{1}}}}+{B^{{H_{2}}}}$ are self-similar. We compare the self-similarity exponents, orders of the Hölder continuity on a finite interval, and exponents ${\lambda _{1}}$ and ${\lambda _{2}}$ in the asymptotics $\operatorname{\mathsf{E}}{X_{1}}{X_{t}}\asymp {t^{{\lambda _{1}}}}$ and $\operatorname{\mathsf{E}}{X_{1}}\hspace{0.1667em}({X_{t+1}}-{X_{t}})\asymp {t^{{\lambda _{2}}}}$ as $t\to +\infty $. The results are shown in Table 1. The process X defined in (1) is a fractional Brownian motion for $\alpha =\frac{1}{2}-\alpha $, $\beta =H-\frac{1}{2}$, $\gamma =H-\frac{3}{2}$ and $C(\alpha ,\beta ,\gamma )=(H-\frac{1}{2}){c_{H}}$, $H\in \big(\frac{1}{2},1\big)$. Otherwise, the process X does not coincide with other processes mentioned in Table 1.
Table 1.
Self-similarity exponents, “waning memory” exponents and maximum order for the Hölder condition for some well-known Gaussian processes
| The process | The self-similarity exponent | The exponent in asymptotics | Hölder condition, up to order | |
| of $\operatorname{\mathsf{E}}{X_{1}}{X_{t}}$ | of $\operatorname{\mathsf{E}}{X_{1}}({X_{t+1}}-{X_{t}})$ | |||
| ${\lambda _{1}}$ | ${\lambda _{2}}$ | |||
| Standard fBm, ${B^{H}}$ | H | $0\vee (2H-1)$ | $2H-2$ if $H\ne \frac{1}{2}$ | H |
| Sub-fractional Brownian motion $({B_{t}^{H}}-{B_{-t}^{H}})/\sqrt{2}$ | H | $2H-1$ | $2H-2$ if $H\ne \frac{1}{2}$ | H |
| Bifractional Brownian motion ${B^{H,K}}$ | $HK$ | $\max (2HK-1$, $2H(K-1))$ |
$2HK-2H-1$ or $2HK-2$ |
$HK$ if $H\in (0,1)$ and $K\in (0,1]$ |
| Mixed fBm ${B_{t}^{{H_{1}}}}+{B_{t}^{{H_{2}}}}$, ${H_{1}}<{H_{2}}$ | not self-similar | $0\vee (2{H_{2}}-1)$ | $2{H_{2}}-2$ | ${H_{1}}$ |
| Process X defined in (1) | $\alpha +\beta +\gamma +\frac{3}{2}$ | $\beta +\gamma +1$ if $\beta +\gamma \ne -1$ | $\beta +\gamma $ | $\min \big(1,\gamma +\frac{3}{2},\alpha +\beta +\gamma +\frac{3}{2}\big)$ on $[0,T]$; $\big(\gamma +\frac{3}{2}\big)\wedge 1$ on $[{t_{0}},T]$. |
In the present paper we are going to prove that the process X has a modification that satisfies the Hölder condition, and to find the upper bound for its order. To that end, we study the asymptotics of the variances of increments of the process X, construct bounds for them, and obtain the so-called generalized quasi-helix property.
For the technical simplicity, in what follows we put $C(\alpha ,\beta ,\gamma )=1$ and consider a process of the form
Here is a small remark on the notation. We adopt common definitions of asymptotic equivalence and negligibility. Notation $f(t)\sim g(t)$ means that $f(t)={c_{1}}(t)g(t)$ for some function ${c_{1}}(t)\to 1$, while $f(t)=o(g(t))$ means that $f(t)={c_{0}}(t)g(t)$ for some function ${c_{0}}(t)\to 0$.
The paper is organized as follows. In Section 2 we prove that, under conditions (2), the process (1) is well-defined and self-similar. In Section 3 we study asymptotic properties of variances and covariances of the increments of the process X. In Section 4 we find the set of parameters for which the process X has stationary increments. Quasi-helix properties of the process X are studied in Section 5; the continuity and the Hölder condition are proved in Section 6. Auxiliary results are obtained in Appendix (Section A).
2 Existence and self-similarity of Gaussian Volterra processes with power-type kernels
2.1 Well-posedness of the process X
For the process defined in (4), the Volterra kernel equals
Therefore,
Thus, the function $K(t,s)$ is homogeneous of degree $\alpha +\beta +\gamma +1$.
Proof.
For any fixed $t>0$, function $K(t,s)$ is continuous in s on $(0,t]$. Let us apply Lemma 4 and consider three cases.
Case 1. If $\beta +\gamma <-1$, then due to Lemma 4
\[ {\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\sim C{s^{\beta +\gamma +1}}\hspace{1em}\text{as}\hspace{2.5pt}s\to 0+\hspace{2.5pt},\]
whence
as $s\to 0+$, where $C>0$ is a constant. Relations (2) imply that $\alpha +\beta +\gamma +1>-\frac{1}{2}$ whence ${\textstyle\int _{0}^{t}}K{(t,s)^{2}}\hspace{0.1667em}ds<\infty $.Case 2. Let $\beta +\gamma =-1$, then due to Lemma 4
\[ {\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\sim \ln (t/s)\hspace{1em}\text{as}\hspace{2.5pt}s\to 0+\hspace{2.5pt},\]
whence
as $s\to 0+$ because $\frac{\alpha -1}{3}<\alpha $. Taking into account that $\frac{\alpha -1}{3}>-\frac{1}{2}$, we get that ${\textstyle\int _{0}^{t}}K{(t,s)^{2}}\hspace{0.1667em}ds<\infty $.Case 3. If $\beta +\gamma >-1$, then due to Lemma 4
\[ {\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\to C{t^{\beta +\gamma +1}}\hspace{1em}\text{as}\hspace{2.5pt}s\to 0+\text{,}\]
whence
as $s\to 0+$. Since $\alpha >-\frac{1}{2}$, we get that ${\textstyle\int _{0}^{t}}K{(t,s)^{2}}\hspace{0.1667em}ds<\infty $.In either case, (6) holds true. Indeed, due to (5),
\[\begin{aligned}{}{\int _{0}^{t}}K{(t,s)^{2}}\hspace{0.1667em}ds& =\frac{{t^{2\alpha +2\beta +2\gamma +2}}}{{T^{2\alpha +2\beta +2\gamma +2}}}{\int _{0}^{t}}K\hspace{-0.1667em}{\left(T,\frac{Ts}{t}\right)^{2}}ds\\ {} & =\frac{{t^{2\alpha +2\beta +2\gamma +3}}}{{T^{2\alpha +2\beta +2\gamma +3}}}{\int _{0}^{T}}K{(T,u)^{2}}\hspace{0.1667em}du\end{aligned}\]
with $2\alpha +2\beta +2\gamma +3>0$. Hence, the supremum in (6) is attained for $t=T$ and the inequality in (6) holds true. Due to (6), the process X in (1) is well-defined and has the bounded variance. □2.2 Self-similarity of the process X
Proof.
According to (5), the covariance function of X is self-similar in the sense that
Notice that the process X is zero-mean and Gaussian. Together with (7), it implies that the process X is self-similar with exponent H. □
(7)
\[\begin{aligned}{}\operatorname{cov}({X_{ks}},{X_{kt}})& ={\int _{0}^{\min (ks,kt)}}K(kt,u)K(ks,u)\hspace{0.1667em}du\\ {} & =k{\int _{0}^{\min (s,t)}}K(kt,tv)K(ks,tv)\hspace{0.1667em}dv\\ {} & ={k^{2H}}{\int _{0}^{\min (s,t)}}K(t,v)K(s,v)\hspace{0.1667em}dv={k^{2H}}\operatorname{cov}({X_{s}},{X_{t}}).\end{aligned}\]3 Asymptotic properties of incremental variances
Then its increments can be represented as
Thus, the variance of the increment is equal to
where $D=\{(s,u,v)\in {\mathbb{R}^{3}}:0<s\le \max ({t_{1}},s)<\min (u,v)\le \max (u,v)\le {t_{2}}\}$. The set D has a mirror symmetry, and the integrand on the right-hand side of (9) is a symmetric function under the permutation of u and v. Therefore, the integrals over two symmetric to each other halves of the set D are equal:
(8)
\[\begin{aligned}{}{X_{{t_{2}}}}-{X_{{t_{1}}}}& ={\int _{0}^{{t_{1}}}}K({t_{2}},s)\hspace{0.1667em}d{W_{s}}+{\int _{{t_{1}}}^{{t_{2}}}}K({t_{2}},s)\hspace{0.1667em}d{W_{s}}-{\int _{0}^{{t_{1}}}}K({t_{1}},s)\hspace{0.1667em}d{W_{s}}\\ {} & ={\int _{0}^{{t_{1}}}}(K({t_{2}},s)-K({t_{1}},s))\hspace{0.1667em}d{W_{s}}+{\int _{{t_{1}}}^{{t_{2}}}}K({t_{2}},s)\hspace{0.1667em}d{W_{s}}\\ {} & ={\int _{0}^{{t_{1}}}}\hspace{-0.1667em}\hspace{-0.1667em}{s^{\alpha }}{\int _{{t_{1}}}^{{t_{2}}}}\hspace{-0.1667em}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\hspace{0.1667em}d{W_{s}}+{\int _{{t_{1}}}^{{t_{2}}}}\hspace{-0.1667em}\hspace{-0.1667em}{s^{\alpha }}{\int _{s}^{{t_{2}}}}\hspace{-0.1667em}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\hspace{0.1667em}d{W_{s}}\\ {} & ={\int _{0}^{{t_{2}}}}{s^{\alpha }}{\int _{\max (s,{t_{1}})}^{{t_{2}}}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\hspace{0.1667em}d{W_{s}},\hspace{2em}0\le {t_{1}}<{t_{2}}.\end{aligned}\](9)
\[\begin{aligned}{}\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})& ={\int _{0}^{{t_{2}}}}{s^{2\alpha }}{\left({\int _{\max (s,{t_{1}})}^{{t_{2}}}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\right)^{\hspace{-0.1667em}2}}\hspace{0.1667em}ds\\ {} & =\underset{D}{\iiint }{s^{2\alpha }}{u^{\beta }}{(u-s)^{\gamma }}{v^{\beta }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}du\hspace{0.1667em}dv>0,\end{aligned}\]
\[\begin{aligned}{}& \underset{\{(s,u,v)\in D\hspace{0.1667em}:\hspace{0.1667em}u\le v\}}{\iiint }{s^{2\alpha }}{u^{\beta }}{(u-s)^{\gamma }}{v^{\beta }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}du\hspace{0.1667em}dv\\ {} & \hspace{2em}=\underset{\{(s,u,v)\in D\hspace{0.1667em}:\hspace{0.1667em}u\ge v\}}{\iiint }{s^{2\alpha }}{u^{\beta }}{(u-s)^{\gamma }}{v^{\beta }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}du\hspace{0.1667em}dv.\end{aligned}\]
Hence,
(10)
\[\begin{aligned}{}\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})& =2\underset{\{(s,u,v)\in D\hspace{0.1667em}:\hspace{0.1667em}u\le v\}}{\iiint }{s^{2\alpha }}{u^{\beta }}{(u-s)^{\gamma }}{v^{\beta }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}du\hspace{0.1667em}dv\\ {} & =2{\int _{{t_{1}}}^{{t_{2}}}}{u^{\beta }}{\int _{u}^{{t_{2}}}}{v^{\beta }}{\int _{0}^{u}}{s^{2\alpha }}{(u-s)^{\gamma }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du.\end{aligned}\]Proposition 2.
Let the process X admit representation (4), where α, β and γ satisfy relations (2). Then for ${t_{0}}>0$ the asymptotic behavior of $\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})$ as $({t_{1}},{t_{2}})\to ({t_{0}},{t_{0}})$ is as follows:
(11)
\[\begin{array}{r@{\hskip0pt}l@{\hskip0pt}r@{\hskip0pt}l}& \displaystyle \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\sim \frac{{t_{0}^{2\alpha +2\beta }}\hspace{0.1667em}|{t_{2}}-{t_{1}}{|^{2\gamma +3}}\hspace{0.1667em}\mathrm{B}(\gamma +1,\hspace{0.1667em}-2\gamma -1)}{(\gamma +1)\hspace{0.1667em}(2\gamma +3)}\hspace{2em}& & \displaystyle \textit{if}\hspace{2.5pt}\gamma <-\frac{1}{2}\textit{,}\\ {} & \displaystyle \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\sim {t_{0}^{2\alpha +2\beta }}\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2}}\hspace{0.1667em}\ln \hspace{-0.1667em}\left(\frac{{t_{0}}}{|{t_{2}}-{t_{1}}|}\right)\hspace{2em}& & \displaystyle \textit{if}\hspace{2.5pt}\gamma =-\frac{1}{2}\textit{,}\\ {} & \displaystyle \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\sim {t_{0}^{2\alpha +2\beta +2\gamma +1}}\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2}}\hspace{0.1667em}\mathrm{B}(2\alpha +1,\hspace{0.1667em}2\gamma +1)\hspace{2em}& & \displaystyle \textit{if}\hspace{2.5pt}\gamma >-\frac{1}{2}\textit{.}\end{array}\]Proof.
Without loss of generality, assume that $0<{t_{1}}<{t_{2}}$. Consider three cases.
Case 1. Let $\gamma <-\frac{1}{2}$. Due to (10),
According to Lemma 5,
(12)
\[\begin{aligned}{}\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})& =2{\int _{{t_{1}}}^{{t_{2}}}}{u^{\beta }}{\int _{u}^{{t_{2}}}}{v^{\beta }}{\int _{0}^{u}}{s^{2\alpha }}{(u-s)^{\gamma }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & =2{\int _{{t_{1}}}^{{t_{2}}}}{\int _{u}^{{t_{2}}}}{u^{\beta }}{v^{\beta }}{\int _{0}^{u}}({s^{2\alpha }}-{u^{2\alpha }})\hspace{0.1667em}{(u-s)^{\gamma }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & +2{\int _{{t_{1}}}^{{t_{2}}}}{u^{2\alpha +\beta }}{\int _{u}^{{t_{2}}}}{v^{\beta }}{\int _{0}^{u}}{(u-s)^{\gamma }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du.\end{aligned}\]
\[\begin{aligned}{}& \underset{\substack{(u,v)\to ({t_{0}},{t_{0}})\\ {} u<v}}{\lim }{u^{\beta }}{v^{\beta }}{\int _{0}^{u}}({s^{2\alpha }}-{u^{2\alpha }})\hspace{0.1667em}{(u-s)^{\gamma }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\\ {} & \hspace{2em}={t_{0}^{2\beta }}{\int _{0}^{{t_{0}}}}({s^{2\alpha }}-{t_{0}^{2\alpha }})\hspace{0.1667em}{({t_{0}}-s)^{2\gamma }}\hspace{0.1667em}ds,\end{aligned}\]
where the integral on the right-hand side is finite.Therefore,
as $({t_{1}},{t_{2}})\to ({t_{0}},{t_{0}})$.
(13)
\[\begin{aligned}{}& 2{\int _{{t_{1}}}^{{t_{2}}}}{\int _{u}^{{t_{2}}}}{u^{\beta }}{v^{\beta }}{\int _{0}^{u}}({s^{2\alpha }}-{u^{2\alpha }})\hspace{0.1667em}{(u-s)^{\gamma }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & \hspace{2em}\sim 2{\int _{{t_{1}}}^{{t_{2}}}}{\int _{u}^{{t_{2}}}}dv\hspace{0.1667em}du\hspace{0.1667em}{t_{0}^{2\beta }}{\int _{0}^{{t_{0}}}}({s^{2\alpha }}-{t_{0}^{2\alpha }})\hspace{0.1667em}{({t_{0}}-s)^{2\gamma }}\hspace{0.1667em}ds\\ {} & \hspace{2em}={({t_{2}}-{t_{1}})^{2}}\hspace{0.1667em}{t_{0}^{2\beta }}{\int _{0}^{{t_{0}}}}({s^{2\alpha }}-{t_{0}^{2\alpha }})\hspace{0.1667em}{({t_{0}}-s)^{2\gamma }}\hspace{0.1667em}ds\end{aligned}\]With Lemma 2.2, (ii) from [7], we come to
as $({t_{1}},{t_{2}})\to ({t_{0}},{t_{0}})$.
\[\begin{aligned}{}& 2{\int _{{t_{1}}}^{{t_{2}}}}{u^{2\alpha +\beta }}{\int _{u}^{{t_{2}}}}{v^{\beta }}{\int _{0}^{u}}{(u-s)^{\gamma }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & \hspace{2em}=2{\int _{{t_{1}}}^{{t_{2}}}}{u^{2\alpha +\beta }}{\int _{u}^{{t_{2}}}}{v^{\beta }}{(v-u)^{2\gamma +1}}{\int _{1}^{v/(v-u)}}{(t-1)^{\gamma }}{t^{\gamma }}\hspace{0.1667em}dt\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & \hspace{2em}=2{\int _{{t_{1}}}^{{t_{2}}}}{u^{2\alpha +\beta }}{\int _{u}^{{t_{2}}}}{v^{\beta }}{(v-u)^{2\gamma +1}}{\int _{0}^{u/v}}{s^{\gamma }}{(1-s)^{-2\gamma -2}}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du.\end{aligned}\]
Since
\[ \underset{\substack{(u,v)\to ({t_{0}},{t_{0}})\\ {} {t_{1}}<{t_{2}}}}{\lim }{u^{2\alpha +\beta }}{v^{\beta }}{\int _{0}^{u/v}}{s^{\gamma }}{(1-s)^{-2\gamma -2}}\hspace{0.1667em}ds={t_{0}^{2\alpha +2\beta }}\mathrm{B}(\gamma +1,\hspace{0.1667em}-2\gamma -1),\]
(here we use the condition $\gamma <-\frac{1}{2}$),
(14)
\[\begin{aligned}{}& 2{\int _{{t_{1}}}^{{t_{2}}}}{u^{2\alpha +\beta }}{\int _{u}^{{t_{2}}}}{v^{\beta }}{\int _{0}^{u}}{(u-s)^{\gamma }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & \hspace{2em}\sim 2{\int _{{t_{1}}}^{{t_{2}}}}{\int _{u}^{{t_{2}}}}{(v-u)^{2\gamma +1}}\hspace{0.1667em}dv\hspace{0.1667em}du\hspace{0.1667em}{t_{0}^{2\alpha +2\beta }}\mathrm{B}(\gamma +1,\hspace{0.1667em}-2\gamma -1)\\ {} & \hspace{2em}=\frac{{({t_{2}}-{t_{1}})^{2\gamma +3}}\hspace{0.1667em}{t_{0}^{2\alpha +2\beta }}\hspace{0.1667em}\mathrm{B}(\gamma +1,\hspace{0.1667em}-2\gamma -1)}{(\gamma +1)\hspace{0.1667em}(2\gamma +3)}\end{aligned}\]The right-hand side of (13) is negligible comparing to the right-hand side of (14). Hence, according to (12), (13), (14),
\[ \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{2}}}})\sim \frac{{({t_{2}}-{t_{1}})^{2\gamma +3}}\hspace{0.1667em}{t_{0}^{2\alpha +2\beta }}\hspace{0.1667em}\mathrm{B}(\gamma +1,\hspace{0.1667em}-2\gamma -1)}{(\gamma +1)\hspace{0.1667em}(2\gamma +3)}\]
as $({t_{1}},{t_{2}})\to ({t_{0}},{t_{0}})$.Case 2. Let $\gamma =-\frac{1}{2}$. Relations (12) and (13) still hold true:
as $({t_{1}},{t_{2}})\to ({t_{0}},{t_{0}})$. It is easy to see that
as $({t_{1}},{t_{2}})\to ({t_{0}},{t_{0}})$, and, according to Lemma 6,
as $({t_{1}},{t_{2}})\to ({t_{0}},{t_{0}})$. Equations (16), (17) and (18) imply that
(15)
\[\begin{aligned}{}& \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\\ {} & \hspace{2em}=2{\int _{{t_{1}}}^{{t_{2}}}}{\int _{u}^{{t_{2}}}}{u^{\beta }}{v^{\beta }}{\int _{0}^{u}}({s^{2\alpha }}-{u^{2\alpha }})\hspace{0.1667em}{(u-s)^{-1/2}}{(v-s)^{-1/2}}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & \hspace{2em}+2{\int _{{t_{1}}}^{{t_{2}}}}{u^{2\alpha +\beta }}{\int _{u}^{{t_{2}}}}{v^{\beta }}{\int _{0}^{u}}{(u-s)^{-1/2}}{(v-s)^{-1/2}}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du,\end{aligned}\](16)
\[\begin{aligned}{}& 2{\int _{{t_{1}}}^{{t_{2}}}}{\int _{u}^{{t_{2}}}}{u^{\beta }}{v^{\beta }}{\int _{0}^{u}}({s^{2\alpha }}-{u^{2\alpha }})\hspace{0.1667em}{(u-s)^{-1/2}}{(v-s)^{-1/2}}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & \hspace{2em}\sim 2\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2}}\hspace{0.1667em}{t_{0}^{2\beta }}{\int _{0}^{{t_{0}}}}\frac{{s^{2\alpha }}-{t_{0}^{2\alpha }}}{{t_{0}}-s}\hspace{0.1667em}ds\end{aligned}\](17)
\[\begin{aligned}{}& {\int _{{t_{1}}}^{{t_{2}}}}{u^{2\alpha +\beta }}{\int _{u}^{{t_{2}}}}{v^{\beta }}{\int _{0}^{u}}{(u-s)^{-1/2}}{(v-s)^{-1/2}}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & \hspace{2em}\sim {t_{0}^{2\alpha +2\beta }}{\int _{{t_{1}}}^{{t_{2}}}}{\int _{u}^{{t_{2}}}}{\int _{0}^{u}}{(u-s)^{-1/2}}{(v-s)^{-1/2}}ds\hspace{0.1667em}dv\hspace{0.1667em}du\end{aligned}\](18)
\[\begin{aligned}{}& {\int _{{t_{1}}}^{{t_{2}}}}{\int _{u}^{{t_{2}}}}{\int _{0}^{u}}{(u-s)^{-1/2}}{(v-s)^{-1/2}}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & \hspace{2em}={t_{2}^{2}}-({t_{1}}+{t_{2}}){t_{1}^{1/2}}{t_{2}^{1/2}}+{t_{1}^{2}}+\frac{{({t_{2}}-{t_{1}})^{2}}}{2}\ln \hspace{-0.1667em}\left(\frac{{t_{2}^{1/2}}+{t_{1}^{1/2}}}{{t_{2}^{1/2}}-{t_{1}^{1/2}}}\right)\\ {} & \hspace{2em}\sim \frac{{({t_{2}}-{t_{1}})^{2}}}{2}\ln \hspace{-0.1667em}\left(\frac{{t_{0}}}{{t_{2}}-{t_{1}}}\right)\end{aligned}\]
\[\begin{aligned}{}& 2{\int _{{t_{1}}}^{{t_{2}}}}{u^{2\alpha +\beta }}{\int _{u}^{{t_{2}}}}{v^{\beta }}{\int _{0}^{u}}{(u-s)^{-1/2}}{(v-s)^{-1/2}}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & \hspace{2em}\sim {t_{0}^{2\alpha +2\beta }}{({t_{2}}-{t_{1}})^{2}}\ln \hspace{-0.1667em}\left(\frac{{t_{0}}}{{t_{2}}-{t_{1}}}\right)\end{aligned}\]
as $({t_{1}},{t_{2}})\to ({t_{0}},{t_{0}})$.Comparing asymptotics of the summands on the right-hand side of (15), we get that the first one is negligible. Thus,
\[ \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\sim {t_{0}^{2\alpha +2\beta }}{({t_{2}}-{t_{1}})^{2}}\ln \hspace{-0.1667em}\left(\frac{{t_{0}}}{{t_{2}}-{t_{1}}}\right)\]
as $({t_{1}},{t_{2}})\to ({t_{0}},{t_{0}})$.Case 3. $\gamma >-\frac{1}{2}$. According to Lemma 7,
\[\begin{aligned}{}\underset{\substack{(u,v)\to ({t_{0}},{t_{0}})\\ {} u<v}}{\lim }{u^{\beta }}{v^{\beta }}{\int _{0}^{u}}{s^{2\alpha }}\hspace{0.1667em}{(u-s)^{\gamma }}{(v-s)^{\gamma }}\hspace{0.1667em}ds& ={t_{0}^{2\beta }}{\int _{0}^{{t_{0}}}}{s^{2\alpha }}\hspace{0.1667em}{({t_{0}}-s)^{2\gamma }}\hspace{0.1667em}ds\\ {} & ={t_{0}^{2\alpha +2\beta +2\gamma +1}}\hspace{0.1667em}\mathrm{B}(2\alpha +1,\hspace{0.1667em}2\gamma +1).\end{aligned}\]
Hence,
\[\begin{aligned}{}\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})& =2{\int _{{t_{1}}}^{{t_{2}}}}{\int _{u}^{{t_{2}}}}{u^{\beta }}{v^{\beta }}{\int _{0}^{u}}{s^{2\alpha }}{(u-s)^{\gamma }}{(v-s)^{\gamma }}\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\ {} & \sim 2{\int _{{t_{1}}}^{{t_{2}}}}{\int _{u}^{{t_{2}}}}\hspace{0.1667em}dv\hspace{0.1667em}du\hspace{0.1667em}{t_{0}^{2\alpha +2\beta +2\gamma +1}}\hspace{0.1667em}\mathrm{B}(2\alpha +1,\hspace{0.1667em}2\gamma +1)\\ {} & ={({t_{2}}-{t_{1}})^{2}}\hspace{0.1667em}{t_{0}^{2\alpha +2\beta +2\gamma +1}}\hspace{0.1667em}\mathrm{B}(2\alpha +1,\hspace{0.1667em}2\gamma +1)\end{aligned}\]
as $({t_{1}},{t_{2}})\to ({t_{0}},{t_{0}})$. □Proposition 3.
Let the process X admit representation (4) with the values of powers satisfying relations (2). Let $0<{t_{2}}<{t_{3}}$. Then the asymptotic behavior of $\operatorname{\mathsf{E}}[{X_{{t_{1}}}}\hspace{0.1667em}({X_{{t_{3}}}}-{X_{{t_{2}}}})]$ as ${t_{1}}\to 0+$ is as follows:
\[\begin{aligned}{}& \operatorname{\mathsf{E}}[{X_{{t_{1}}}}\hspace{0.1667em}({X_{{t_{3}}}}-{X_{{t_{2}}}})]\\ {} & \hspace{2em}\sim \frac{\mathrm{B}(2\alpha +1,\hspace{0.2222em}\gamma +1)}{(2\alpha +\beta +\gamma +2)(\beta +\gamma +1)}\hspace{0.1667em}({t_{3}^{\beta +\gamma +1}}-{t_{2}^{\beta +\gamma +1}})\hspace{0.1667em}{t_{1}^{2\alpha +\beta +\gamma +2}}\end{aligned}\]
if $\beta +\gamma \ne -1$,
\[ \operatorname{\mathsf{E}}[{X_{{t_{1}}}}\hspace{0.1667em}({X_{{t_{3}}}}-{X_{{t_{2}}}})]\sim \frac{\mathrm{B}(2\alpha +1,\hspace{0.2222em}\gamma +1)}{2\alpha +\beta +\gamma +2}\hspace{0.1667em}{t_{1}^{2\alpha +\beta +\gamma +2}}\hspace{0.1667em}\ln \hspace{-0.1667em}\left(\frac{{t_{3}}}{{t_{2}}}\right)\]
if $\beta +\gamma =-1$.
Proof.
According to (4) and (8), for $0<{t_{1}}<{t_{2}}<{t_{3}}$
\[ \operatorname{\mathsf{E}}[{X_{{t_{1}}}}\hspace{0.1667em}({X_{{t_{3}}}}-{X_{{t_{2}}}})]={\int _{0}^{{t_{1}}}}{s^{2\alpha }}\bigg({\int _{s}^{{t_{1}}}}{u^{\beta }}\hspace{0.1667em}{(u-s)^{\gamma }}\hspace{0.1667em}du\bigg)\hspace{0.1667em}\bigg({\int _{{t_{2}}}^{{t_{3}}}}{v^{\beta }}\hspace{0.1667em}{(v-s)^{\gamma }}\hspace{0.1667em}dv\bigg)\hspace{0.1667em}ds.\]
Obviously,
Furthermore,
since the assumptions (2) ensure that $2\alpha +\beta +\gamma +2>0$. By (19) and (20),
\[ \underset{s\to 0}{\lim }{\int _{{t_{2}}}^{{t_{3}}}}{v^{\beta }}\hspace{0.1667em}{(v-s)^{\gamma }}\hspace{0.1667em}dv={\int _{{t_{2}}}^{{t_{3}}}}{v^{\beta +\gamma }}\hspace{0.1667em}dv=C({t_{2}},{t_{3}}),\]
where
\[ C({t_{2}},{t_{3}})=\left\{\begin{array}{l@{\hskip10.0pt}l}\displaystyle \frac{{t_{3}^{\beta +\gamma +1}}-{t_{2}^{\beta +\gamma +1}}}{\beta +\gamma +1}\hspace{1em}& \text{if}\hspace{2.5pt}\beta +\gamma \ne -1,\\ {} \ln ({t_{3}}/{t_{2}})\hspace{1em}& \text{if}\hspace{2.5pt}\beta +\gamma =-1.\end{array}\right.\]
Hence,
(19)
\[ \operatorname{\mathsf{E}}[{X_{{t_{1}}}}\hspace{0.1667em}({X_{{t_{3}}}}-{X_{{t_{2}}}})]\sim C({t_{2}},{t_{3}}){\int _{0}^{{t_{1}}}}{s^{2\alpha }}\bigg({\int _{s}^{t}}{u^{\beta }}\hspace{0.1667em}{(u-s)^{\gamma }}\hspace{0.1667em}du\bigg)\hspace{0.1667em}ds\hspace{1em}\text{as}\hspace{2.5pt}{t_{1}}\to 0\text{.}\](20)
\[\begin{aligned}{}{\int _{0}^{{t_{1}}}}{s^{2\alpha }}\bigg({\int _{s}^{t}}{u^{\beta }}\hspace{0.1667em}{(u-s)^{\gamma }}\hspace{0.1667em}du\bigg)\hspace{0.1667em}ds& ={\int _{0}^{{t_{1}}}}{u^{\beta }}\bigg({\int _{0}^{u}}{s^{2\alpha }}\hspace{0.1667em}{(u-s)^{\gamma }}\hspace{0.1667em}ds\bigg)\hspace{0.1667em}du\\ {} & =\mathrm{B}(2\alpha +1,\hspace{0.2222em}\gamma +1){\int _{0}^{{t_{1}}}}{u^{2\alpha +\beta +\gamma +1}}\hspace{0.1667em}\hspace{0.1667em}du\\ {} & =\frac{\mathrm{B}(2\alpha +1,\hspace{0.2222em}\gamma +1)}{2\alpha +\beta +\gamma +2}\hspace{0.1667em}{t_{1}^{2\alpha +\beta +\gamma +2}},\end{aligned}\]
\[ \operatorname{\mathsf{E}}[{X_{{t_{1}}}}\hspace{0.1667em}({X_{{t_{3}}}}-{X_{{t_{2}}}})]\sim \frac{C({t_{2}},{t_{3}})\hspace{0.1667em}\mathrm{B}(2\alpha +1,\hspace{0.2222em}\gamma +1)}{2\alpha +\beta +\gamma +2}\hspace{0.1667em}{t_{1}^{2\alpha +\beta +\gamma +2}}\hspace{1em}\text{as}\hspace{2.5pt}{t_{1}}\to 0,\]
as desired. □4 When does the process $\boldsymbol{X}$ have stationary increments?
Recall that fractional Brownian motion with Hurst index $H\in (0,1)$ is a zero-mean Gaussian process with covariance function $\operatorname{cov}({X_{s}},{X_{t}})=({s^{2H}}+{t^{2H}}-|t-s{|^{2H}})/2$.
Theorem 2.
Let stochastic process X be defined by relations (4) and (2). Then the following three statements are equivalent:
-
(a) The process X has stationary increments.
-
(b) Up to a constant, the process X is a fractional Brownian motion with Hurst index $H\in \big(\frac{1}{2},1\big)\hspace{0.1667em}$.
-
(c) There exists $H\in \big(\frac{1}{2},1\big)$ such that $\alpha =\frac{1}{2}-H$, $\beta =H-\frac{1}{2}$ and $\gamma =H-\frac{3}{2}$.
Proof.
The process X is Gaussian and according to Proposition 1, it is also self-similar with exponent $H=\alpha +\beta +\gamma +\frac{3}{2}$. Suppose $(a)$, i.e., it has stationary increments. According to [3, Section 1.3; Theorem 1.3.1] a self-similar Gaussian process with stationary increments is a fBm, up to a constant. Moreover, $H>0$ and
In cases (ii) and (iii) $\operatorname{var}[{X_{{t_{2}}}}\mid {X_{{t_{1}}}}]=0$ for ${t_{1}}<{t_{2}}$, which contradicts (44). Thus, case (i) takes place, and up to a constant, the process X is a fBm, ${X_{t}}=m{B_{t}^{H}}$ with exponent $H\in (0,1)$. Then $\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})={m^{2}}\hspace{0.1667em}|{t_{2}}-{t_{1}}{|^{2H}}$. On the other hand, the asymptotics of $\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})$ is obtained in Proposition 2. Since $2H<2$, the first case in Proposition 2 occurs, namely, $\gamma <-\frac{1}{2}$ and the asymptotics satisfies (11). It means that
-
(i) if $H\in (0,1)$, then the process X is a fractional Brownian motion with Hurst index H;
-
(ii) if $H=1$, then $X(t)=tX(1)$ a.s. for all $t\ge 0$ and for some Gaussian variable $X(1)$ (see Theorem 1.3.3 in [3]);
\[ {m^{2}}\hspace{0.1667em}|{t_{2}}-{t_{1}}{|^{2H}}\sim {C^{2}}(\alpha ,\beta ,\gamma )\hspace{0.1667em}\frac{{t_{0}^{2\alpha +2\beta }}\hspace{0.1667em}|{t_{2}}-{t_{1}}{|^{2\gamma +3}}\hspace{0.1667em}\mathrm{B}(\gamma +1,\hspace{0.1667em}-2\gamma -1)}{(\gamma +1)\hspace{0.1667em}(2\gamma +3)}\]
as ${t_{1}}\to {t_{0}}$ and ${t_{2}}\to {t_{0}}$, for all ${t_{0}}\in (0,T]$. Equating the exponents, we obtain that $2\alpha +2\beta =0$ and $2\gamma +3=2H$. Since $\gamma \in \big(-1,-\frac{1}{2}\big)$, one has $H\in \big(\frac{1}{2},1\big)$, and we get that $(a)$ implies $(b)$.Having $(b)$, find the asymptotics for $\operatorname{\mathsf{E}}[{X_{{t_{1}}}}\hspace{0.1667em}({X_{{t_{3}}}}-{X_{{t_{2}}}})]$:
\[\begin{aligned}{}\operatorname{\mathsf{E}}[{X_{{t_{1}}}}\hspace{0.1667em}({X_{{t_{3}}}}-{X_{{t_{2}}}})]& =\frac{{m^{2}}\hspace{0.1667em}({t_{3}^{2H}}-|{t_{3}}-{t_{1}}{|^{2H}}-{t_{2}^{2H}}+|{t_{2}}-{t_{1}}{|^{2H}})}{2}\\ {} & \sim H\hspace{0.1667em}{m^{2}}\hspace{0.1667em}({t_{3}^{2H-1}}-{t_{2}^{2H-1}})\hspace{0.1667em}{t_{1}}\end{aligned}\]
as ${t_{1}}\to 0$, for all ${t_{2}}\in (0,T]$ and ${t_{3}}\in (0,T]$ such that ${t_{2}}\ne {t_{3}}$. Compare this with the result of Proposition 3. The first case, $\beta +\gamma \ne -1$, occurs in Proposition 3, and
\[ H\hspace{0.1667em}{m^{2}}\hspace{0.1667em}({t_{3}^{2H-1}}-{t_{2}^{2H-1}})\hspace{0.1667em}{t_{1}}\sim C({t_{3}^{\beta +\gamma +1}}-{t_{2}^{\beta +\gamma +1}})\hspace{0.1667em}{t_{1}^{2\alpha +\beta +\gamma +2}}\]
as ${t_{1}}\to 0$, where $C>0$ is a constant. Thus, $\beta +\gamma +1=2H-1$ and $2\alpha +\beta +\gamma +2=1$. Now we can find α, β and γ from the system of linear equations:
\[ 2\alpha +2\beta =0,\hspace{1em}2\gamma +3=2H,\hspace{1em}\beta +\gamma +1=2H-1,\hspace{1em}2\alpha +\beta +\gamma +2=1,\]
whence
So, $(b)$ implies $(c)$. Implication $(c)\Rightarrow (a)$ is evident. □5 Quasi-helix and generalized quasi-helix conditions
In this section we present the uniform inequalities for the incremental variances of Gaussian processes with Volterra kernels.
5.1 Definitions
Definition 1.
Let $0\le {t_{0}}<T$. The process $\{{X_{t}},\hspace{0.2778em}t\in [{t_{0}},T]\}$ is a quasi-helix with exponent $\lambda >0$ if there exist two constants ${C_{i}}>0$, $i=1,2$, such that for any ${t_{0}}\le {t_{1}}<{t_{2}}\le T$
Sometimes we can construct lower and upper bounds for the variance with different exponents. Thus, we come to the notion of the generalized quasi-helix.
Definition 2.
The process $\{{X_{t}},\hspace{0.2778em}t\in [{t_{0}},T]\}$ is a generalized quasi-helix with exponents ${\lambda _{i}}>0$, $i=1,2$, if there exist two constants ${C_{i}}>0$, $i=1,2$, such that for any ${t_{0}}\le {t_{1}}<{t_{2}}\le T$
Remark 2.
Unless $\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})=0$ for all ${t_{1}},{t_{2}}\in [{t_{0}},T]$, the exponents ${\lambda _{i}}$ satisfy the relation $0<{\lambda _{2}}\le \min (1,{\lambda _{1}})$.
Definition 3.
The process $\{{X_{t}},\hspace{0.2778em}t\in [{t_{0}},T]\}$ is a pseudo-quasihelix with exponent $\lambda >0$ if for any ${\lambda _{1}}$ and ${\lambda _{2}}$ such that $0<{\lambda _{2}}<\lambda <{\lambda _{1}}$ it is a generalized quasi-helix with exponents ${\lambda _{1}}$ and ${\lambda _{2}}$.
5.2 Quasi-helix on $[{t_{0}},T]$
The following lemma allows to figure out when a self-similar process is a quasi-helix considering the asymptotic behavior of its small incremental variances.
Lemma 1.
Let $0<{t_{0}}<T$, and let the stochastic process $\{{X_{t}},\hspace{0.2778em}t\in (0,T]\}$ satisfy the following conditions:
-
(i) $\{{X_{t}},\hspace{0.2778em}t\in (0,T]\}$ is self-similar with exponent H;
-
(ii) $\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})>0$ whenever ${t_{0}}\le {t_{1}}<{t_{2}}\le T$;
-
(iii) for some $C>0$ and $\lambda >0$ and for all $t\in [{t_{0}},T]$ the variances of increments asymptotically behave as follows:
Then the process $\{{X_{t}},\hspace{0.2778em}t\in [{t_{0}},T]\}$ is a quasi-helix with exponent λ.
Proof.
Because of (22), the process $\{{X_{t}},\hspace{0.2778em}t\in [{t_{0}},T]\}$ is mean-square continuous. Now, consider the function
\[ f(t)=\frac{\operatorname{var}({X_{t}}-{X_{{t_{0}}}})}{{t_{0}^{2H-2\lambda }}{(t-{t_{0}})^{2\lambda }}},\hspace{1em}t\in ({t_{0}},T].\]
Obviously, function $f(t)$ is continuous on $({t_{0}},T]$, ${\lim \nolimits_{t\to {t_{0}}}}f(t)=C>0$ and $f(t)>0$, $t\in ({t_{0}},T]$, due to condition (iii). As a consequence, it is bounded on $[{t_{0}},T]$, and there exist $0<{c_{1}}<{c_{2}}$ such that ${c_{1}}\le f(t)\le {c_{2}}$ for all $t\in ({t_{0}},T]$. Since the process X is self-similar with exponent H, we have that
\[ \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})=\frac{{t_{1}^{2H}}}{{t_{0}^{2H}}}\operatorname{var}({X_{{t_{2}}{t_{0}}/{t_{1}}}}-{X_{{t_{0}}}})={t_{1}^{2H-2\lambda }}{({t_{2}}-{t_{1}})^{2\lambda }}f\hspace{-0.1667em}\left(\frac{{t_{2}}{t_{0}}}{{t_{1}}}\right)\]
for all ${t_{1}}$ and ${t_{2}}$ such that ${t_{0}}\le {t_{1}}<{t_{2}}\le T$.Thus,
whence
It means that inequality (21) holds true for ${C_{1}}={c_{1}}\min ({t_{0}^{2H-2\lambda }},{T^{2H-2\lambda }})$ and ${C_{2}}={c_{2}}\max ({t_{0}^{2H-2\lambda }},{T^{2H-2\lambda }})$. So, the process X is a quasi-helix on the interval $[{t_{0}},T]$ with exponent λ. □
Theorem 3.
Let $0<{t_{0}}<T$, and let the process X be defined by (4) and (2). Moreover, assume that $\gamma \ne -\frac{1}{2}$. Then the process $\{{X_{t}},\hspace{0.2778em}t\in [{t_{0}},T]\}$ is a quasi-helix with exponent $\gamma +\frac{3}{2}$ if $-1<\gamma <-\frac{1}{2}$, and with exponent 1 if $\gamma >-\frac{1}{2}$.
5.3 Generalized quasi-helix on $[0,T]$. Case $\gamma \ne -1/2$
Lemma 2.
Let $T>0$ and let stochastic process $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ satisfy the following conditions:
Then the process $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ is a generalized quasi-helix with exponents $H\vee \lambda $ and $H\wedge \lambda $.
-
(i) ${X_{0}}=0$;
-
(ii) $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ is self-similar with exponent $H>0$;
-
(iii) $\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})>0$ whenever $0\le {t_{1}}<{t_{2}}\le T$;
-
(iv) for some $C>0$, $\lambda >0$ and for all $t\in (0,T]$ the incremental variances asymptotically behave as follows:
Proof.
The process $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ is mean-square continuous. The continuity at point 0 follows from self-similarity with exponent $H>0$, while the continuity on $(0,T]$ follows from (23) with $\lambda >0$.
Consider the function
The function $f(t)$ is continuous on $[0,T)$, $f(t)>0$ on $[0,T)$, and
As the consequence, there exist ${c_{1}}>0$ and ${c_{2}}$ such that ${c_{1}}\le f(t)\le {c_{2}}$ on $[0,T)$.
Because of self-similarity, for any $0\le {t_{1}}<{t_{2}}\le T$
\[ \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})=\frac{{t_{2}^{2H}}}{{T^{2H}}}\operatorname{var}({X_{T}}-{X_{{t_{1}}T/{t_{2}}}})={t_{2}^{2H-2\lambda }}{T^{2\lambda -2H}}{({t_{2}}-{t_{1}})^{2\lambda }}f\hspace{-0.1667em}\left(\frac{{t_{1}}T}{{t_{2}}}\right).\]
Hence,
\[ {c_{1}}{t_{2}^{2H-2\lambda }}{T^{2\lambda -2H}}{({t_{2}}-{t_{1}})^{2\lambda }}\le \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\le {c_{2}}{t_{2}^{2H-2\lambda }}{T^{2\lambda -2H}}{({t_{2}}-{t_{1}})^{2\lambda }}.\]
In the following inequalities, we use that $0<{t_{2}}-{t_{1}}\le {t_{2}}\le T$, however the inequalities depend on the sign of $2H-2\lambda $. If $2H\le 2\lambda $, then
\[\begin{aligned}{}{c_{1}}{({t_{2}}-{t_{1}})^{2\lambda }}& \le \frac{{c_{1}}\hspace{0.1667em}{T^{2\lambda -2H}}{({t_{2}}-{t_{1}})^{2\lambda }}}{{t_{2}^{2\lambda -2H}}}\le \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\\ {} & \le \frac{{c_{2}}\hspace{0.1667em}{T^{2\lambda -2H}}{({t_{2}}-{t_{1}})^{2\lambda }}}{{t_{2}^{2\lambda -2H}}}\le {c_{2}}\hspace{0.1667em}{T^{2\lambda -2H}}{({t_{2}}-{t_{1}})^{2H}}.\end{aligned}\]
If $2H\ge 2\lambda $, then
\[\begin{aligned}{}\frac{{c_{1}}{({t_{2}}-{t_{1}})^{2H}}}{{T^{2H-2\lambda }}}& \le \frac{{c_{1}}\hspace{0.1667em}{t_{2}^{2H-2\lambda }}{({t_{2}}-{t_{1}})^{2\lambda }}}{{T^{2H-2\lambda }}}\le \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\\ {} & \le \frac{{c_{2}}\hspace{0.1667em}{t_{2}^{2H-2\lambda }}{({t_{2}}-{t_{1}})^{2\lambda }}}{{T^{2H-2\lambda }}}\le {c_{2}}\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2\lambda }},\end{aligned}\]
and the proof follows. □Theorem 4.
Let $T>0$, and let the process X be defined by (4) and (2). Moreover, assume that $\gamma \ne -\frac{1}{2}$. If $-1<\gamma <-\frac{1}{2}$, then the process $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ is a generalized quasi-helix with exponents $(\gamma +\frac{3}{2})\vee (\alpha +\beta +\gamma +\frac{3}{2})$ and $(\gamma +\frac{3}{2})\wedge (\alpha +\beta +\gamma +\frac{3}{2})$. Otherwise, if $\gamma >-\frac{1}{2}$, then the process $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ is a generalized quasi-helix with exponents $1\vee (\alpha +\beta +\gamma +\frac{3}{2})$ and $1\wedge (\alpha +\beta +\gamma +\frac{3}{2})$.
If a self-similar process is a quasi-helix on $[0,T]$, then the exponents in the self-similarity condition and in the quasi-helix condition must be the same. If for some ${t_{0}}\in [0,T]$ the variances of a quasi-helix satisfy the relation $\operatorname{var}({X_{{t_{1}}}}-{X_{{t_{2}}}})\sim C({t_{0}})\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2\lambda }}$ as ${t_{1}}\to {t_{0}}$, ${t_{2}}\to {t_{0}}$, ${t_{1}}<{t_{2}}$ for some $C({t_{0}})>0$ and $\lambda >0$, then the exponent in the quasi-helix condition must be equal to λ. If the variances of a stochastic process satisfy the relation given in the second case in Proposition 2 (for $\gamma =-\frac{1}{2}$), then the process cannot be quasi-helix. This proves the necessity condition in the following corollary. The sufficiency follows from Theorems 3 and 4.
5.4 Generalized quasi-helix on $[0,T]$. The bordering case $\gamma =-\frac{1}{2}$
Theorem 5.
Let $0<{t_{0}}<T$, and let the process X be defined by (4) and (2) with
Then the following holds true:
(24)
\[ \alpha >-\frac{1}{2},\hspace{2em}\alpha +\beta >-1,\hspace{2em}\textit{and}\hspace{2em}\gamma =-\frac{1}{2}.\]-
1. For any $\lambda \in (0,1)$ the process $\{{X_{t}},\hspace{0.2778em}t\in [{t_{0}},T]\}$ is a generalized quasi-helix with exponents 1 and λ.
-
2. If $\alpha +\beta <0$, then the process $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ is a generalized quasi-helix with exponents 1 and $\alpha +\beta +1$.
-
3. If $\alpha +\beta \ge 0$, then for any $\lambda \in (0,1)$ the process $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ is a generalized quasi-helix with exponents $\alpha +\beta +1$ and λ.
Proof.
We divide the proof into five parts. First, we obtain the lower and upper bounds for $\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})$. Then we subsequently prove the three statements of the theorem.
To start with, the process X is self-similar with the exponent $\alpha +\beta +\gamma +\frac{3}{2}=\alpha +\beta +1>0$. It is square-continuous on $[0,T]$. Denote
\[ {f_{\lambda }}(t)=\frac{\operatorname{var}({X_{T}}-{X_{t}})}{{(T-t)^{2\lambda }}},\hspace{2em}t\in [0,T).\]
Here we assume $\lambda \in \mathbb{R}$. The function ${f_{\lambda }}$ is continuous on $[0,T)$, ${f_{\lambda }}(t)>0$ for all $t\in [0,T)$ and ${f_{\lambda }}(t)\sim {T^{2\alpha +2\beta }}{(T-t)^{2-2\lambda }}(\ln (T)-\ln (T-t))$ as $t\to T$, due to Proposition 2.(i) Lower bound. Let’s apply function ${f_{\lambda }}$ with $\lambda =1$. In this case
\[\begin{array}{l}\displaystyle {f_{1}}(t)=\frac{\operatorname{var}({X_{T}}-{X_{t}})}{{(T-t)^{2}}},\hspace{2em}t\in [0,T),\\ {} \displaystyle {f_{1}}(t)\sim {T^{2\alpha +2\beta }}(\ln (T)-\ln (T-t))\hspace{1em}\text{as}\hspace{1em}t\to T,\\ {} \displaystyle \underset{t\to T}{\lim }{f_{1}}(t)=+\infty .\end{array}\]
Together with continuity and positivity of function ${f_{1}}$ on $[0,T)$ this implies that for some ${c_{1}}>0$
Furthermore, self-similarity of X implies that
\[\begin{aligned}{}\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})& =\frac{{t_{2}^{2\alpha +2\beta +2}}}{{T^{2\alpha +2\beta +2}}}\operatorname{var}({X_{T}}-{X_{{t_{1}}T/{t_{2}}}})\\ {} & =\frac{{t_{2}^{2\alpha +2\beta }}{({t_{2}}-{t_{1}})^{2}}}{{T^{2\alpha +2\beta }}}{f_{1}}\hspace{-0.1667em}\left(\frac{{t_{1}}T}{{t_{2}}}\right)\ge \frac{{c_{1}}\hspace{0.1667em}{t_{2}^{2\alpha +2\beta }}{({t_{2}}-{t_{1}})^{2}}}{{T^{2\alpha +2\beta }}}.\end{aligned}\]
If $\alpha +\beta <0$, then
for all ${t_{1}}$ and ${t_{2}}$ such that $0\le {t_{1}}<{t_{2}}\le T$.
(25)
\[ \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\ge {c_{1}}\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2}}\]If $\alpha +\beta \ge 0$, then
for all $0<{t_{0}}\le {t_{1}}<{t_{2}}\le T$, and
for all $0\le {t_{1}}<{t_{2}}\le T$.
(26)
\[ \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\ge \frac{{c_{1}}\hspace{0.1667em}{t_{0}^{2\alpha +2\beta }}{({t_{2}}-{t_{1}})^{2}}}{{T^{2\alpha +2\beta }}}\](27)
\[ \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\ge \frac{{c_{1}}{({t_{2}}-{t_{1}})^{2\alpha +2\beta +2}}}{{T^{2\alpha +2\beta }}}\](ii) Upper bound. Let $\lambda \in (0,1)$. Then, as $2-2\lambda >0$,
With continuity of ${f_{\lambda }}$ on $[0,T)$, this implies that the function ${f_{\lambda }}$ is bounded on $[0,T)$. Thus, for some finite ${c_{2}}(\lambda )$,
for all ${t_{1}}$ and ${t_{2}}$ such that ${t_{0}}\le {t_{1}}<{t_{2}}\le T$.
\[ \forall \hspace{0.1667em}t\in [0,T)\hspace{0.1667em}:\hspace{0.1667em}{f_{\lambda }}(t)\le {c_{2}}(\lambda ).\]
Furthermore,
\[\begin{aligned}{}\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})& =\frac{{t_{2}^{2\alpha +2\beta +2}}}{{T^{2\alpha +2\beta +2}}}\operatorname{var}({X_{T}}-{X_{{t_{1}}T/{t_{2}}}})\\ {} & =\frac{{t_{2}^{2\alpha +2\beta +2-2\lambda }}{({t_{2}}-{t_{1}})^{2\lambda }}}{{T^{2\alpha +2\beta +2-2\lambda }}}{f_{\lambda }}\hspace{-0.1667em}\left(\frac{{t_{1}}T}{{t_{2}}}\right)\le \frac{{c_{2}}(\lambda )\hspace{0.1667em}{t_{2}^{2\alpha +2\beta +2-2\lambda }}{({t_{2}}-{t_{1}})^{2\lambda }}}{{T^{2\alpha +2\beta +2-2\lambda }}}.\end{aligned}\]
Obviously,
\[ \underset{{t_{2}}\in [{t_{0}},T]}{\max }{t_{2}^{2\alpha +2\beta +2-2\lambda }}=\max ({t_{0}^{2\alpha +2\beta +2-2\lambda }},\hspace{0.2222em}{T^{2\alpha +2\beta +2-2\lambda }}).\]
Thus,
(28)
\[ \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\le {c_{2}}(\lambda )\hspace{0.1667em}\max \hspace{-0.1667em}\left(\frac{{t_{0}^{2\alpha +2\beta +2-2\lambda }}}{{T^{2\alpha +2\beta +2-2\lambda }}},\hspace{0.2222em}1\right){({t_{2}}-{t_{1}})^{2\lambda }}\]If $\lambda \le \alpha +\beta +1$ in addition to $\lambda \in (0,1)$, then
for all ${t_{1}}$ and ${t_{2}}$ such that $0\le {t_{1}}<{t_{2}}\le T$.
(29)
\[ \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\le {c_{2}}(\lambda )\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2\lambda }}\](iii) Proof of statement 1. Let $\lambda \in (0,1)$. If $\alpha +\beta <0$, then, due to (25) and (28),
\[ {c_{1}}\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2}}\le \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\le {c_{2}}(\lambda )\hspace{0.1667em}\max \hspace{-0.1667em}\left(\frac{{t_{0}^{2\alpha +2\beta +2-2\lambda }}}{{T^{2\alpha +2\beta +2-2\lambda }}},\hspace{0.2222em}1\right){({t_{2}}-{t_{1}})^{2\lambda }}.\]
If $\alpha +\beta \ge 0$ (and, as a consequence, $\lambda <\alpha +\beta +1$), then, due to (26) and (29),
\[ \frac{{c_{1}}\hspace{0.1667em}{t_{0}^{2\alpha +2\beta }}{({t_{2}}-{t_{1}})^{2}}}{{T^{2\alpha +2\beta }}}\le \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\le {c_{2}}(\lambda )\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2\lambda }}\]
for all ${t_{1}}$ and ${t_{2}}$ such that ${t_{0}}\le {t_{1}}<{t_{2}}\le T$.In either case, the process $\{{X_{t}},\hspace{0.2778em}t\in [{t_{0}},T]\}$ is a generalized quasi-helix with exponents 1 and λ.
(iv) Proof of statement 2. Let $\alpha +\beta <0$. Denote the self-similarity exponent by λ: $\lambda =\alpha +\beta +1$. Then $\lambda \in (0,1)$. The condition $\lambda \le \alpha +\beta +1$ of (29) is also satisfied.
Due to (25) and (29),
\[ {c_{1}}\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2}}\le \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\le {c_{2}}(\lambda )\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2\lambda }}\]
for all ${t_{1}}$ and ${t_{2}}$ such that $0\le {t_{1}}<{t_{2}}\le T$. Thus, the process $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ is a generalized quasi-helix with exponents 1 and λ. We recall that $\lambda =\alpha +\beta +1$.(v) Proof of statement 3. Let $\alpha +\beta \ge 0$ and $\lambda \in (0,1)$. Then the assumption $\lambda \le \alpha +\beta +1$ of (29) is satisfied. Due to (27) and (29),
\[ \frac{{c_{1}}{({t_{2}}-{t_{1}})^{2\alpha +2\beta +2}}}{{T^{2\alpha +2\beta }}}\le \operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\le {c_{2}}(\lambda )\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2\lambda }}\]
for all ${t_{1}}$ and ${t_{2}}$ such that $0\le {t_{1}}<{t_{2}}\le T$. Thus, the process $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ is a generalized quasi-helix with exponents $\alpha +\beta +1$ and λ. □Quasi-helix, pseudo-quasihelix and generalized quasi-helix conditions for the process X defined by (1) and (2) are summarized in Table 2.
Table 2.
Summary of quasi-helix properties
| The process X is | ||
| $\begin{array}{c}{X_{t}}={\displaystyle\displaystyle \int _{0}^{t}}{s^{\alpha }}{\displaystyle\displaystyle \int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\hspace{0.1667em}d{W_{s}},\\ {} \alpha >-\frac{1}{2},\hspace{2em}\gamma >-1,\hspace{2em}\alpha +\beta +\gamma >-\frac{3}{2}.\end{array}$ | ||
| The process X satisfies | on the interval $[0,T]$ | on any interval $[{t_{0}},T]$ |
| quasi-helix | if and only if $\gamma <-\frac{1}{2}$ and $\alpha +\beta =0$, or $\gamma >-\frac{1}{2}$ and $\alpha +\beta +\gamma =-\frac{1}{2}$. |
$\gamma \ne -\frac{1}{2}$ |
| pseudo-quasihelix |
$\gamma \le -\frac{1}{2}$ and $\alpha +\beta =0$, or $\gamma \ge -\frac{1}{2}$ and $\alpha +\beta +\gamma =-\frac{1}{2}$. |
always |
| generalized quasi-helix | always | always |
| Here $0<{t_{0}}<T$. The entry “always” means “always whenever $\alpha >-\frac{1}{2}$, $\gamma >-1$, $\alpha +\beta +\gamma >-\frac{3}{2}$.” | ||
| The exponents in the generalized quasi-helix condition: | ||
| the exponents in the generalized quasi-helix condition | ||
| on the interval $[0,T]$ are | on any interval $[{t_{0}},T]$ are | |
| If $\gamma <-\frac{1}{2}$ and $\alpha +\beta \le 0$, | $\gamma +\frac{3}{2}$ and $\alpha +\beta +\gamma +\frac{3}{2}$ | $\gamma +\frac{3}{2}$ |
| If $\gamma <-\frac{1}{2}$ and $\alpha +\beta \ge 0$, | $\alpha +\beta +\gamma +\frac{3}{2}$ and $\gamma +\frac{3}{2}$ | $\gamma +\frac{3}{2}$ |
| If $\gamma =-\frac{1}{2}$ and $\alpha +\beta <0$, | 1 and $\alpha +\beta +1$ | 1 and $1-\epsilon $ |
| If $\gamma =-\frac{1}{2}$ and $\alpha +\beta \ge 0$, | $\alpha +\beta +1$ and $1-\epsilon $ | 1 and $1-\epsilon $ |
| If $\gamma >-\frac{1}{2}$ and $\alpha +\beta +\gamma \le -\frac{1}{2}$, | 1 and $\alpha +\beta +\gamma +\frac{3}{2}$ | 1 |
| If $\gamma >-\frac{1}{2}$ and $\alpha +\beta +\gamma \ge -\frac{1}{2}$, | $\alpha +\beta +\gamma +\frac{3}{2}$ and 1 | 1 |
If only one number is in the cell, then the process is a quasi-helix. In that case, the exponents in both sides of the generalized quasi-helix condition are equal. The number $1-\epsilon $ means that the upper bound in the generalized quasi-helix condition “$\exists {C_{2}}\hspace{0.1667em}\forall {t_{1}}\hspace{0.1667em}\forall {t_{2}}:\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\le {C_{2}}\hspace{0.1667em}{({t_{2}}-{t_{1}})^{2\lambda }}$” holds true for all $\lambda \in (0,1)$.
6 Hölder property
The Hölder property for stochastic processes follows from generalized quasi-helix property. Indeed, it is well-known that Gaussian process $\{{X_{t}},\hspace{0.2778em}t\in [{t_{0}},T]\}$ satisfying for some ${\lambda _{0}}>0$ the assumption
\[ \forall \hspace{0.1667em}\lambda \in (0,{\lambda _{0}})\hspace{0.2778em}\exists \hspace{0.1667em}C(\lambda )\hspace{0.2778em}\forall {t_{1}}\in [{t_{0}},T]\hspace{0.2778em}\forall {t_{2}}\in [{t_{0}},T]\hspace{0.1667em}:\hspace{0.1667em}\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})\le C(\lambda )\hspace{0.1667em}|{t_{2}}-{t_{1}}{|^{2\lambda }},\]
has a modification $\widetilde{X}$ whose paths are Hölder up to order ${\lambda _{0}}$, that is,
\[ \forall \hspace{0.1667em}\lambda \in (0,{\lambda _{0}})\hspace{0.2778em}\exists \hspace{0.1667em}C(\lambda ,\omega )\hspace{0.2778em}\forall {t_{1}}\in [{t_{0}},T]\hspace{0.2778em}\forall {t_{2}}\in [{t_{0}},T]\hspace{0.1667em}:\hspace{0.1667em}|{\widetilde{X}_{{t_{2}}}}-{\widetilde{X}_{{t_{1}}}}|\le C(\lambda ,\omega )\hspace{0.1667em}|{t_{2}}-{t_{1}}{|^{\lambda }}.\]
As a consequence, a stochastic process satisfying quasi-helix or pseudo-quasihelix condition with exponent λ also has a modification that is Hölder up to order λ. A stochastic process that satisfies the generalized quasi-helix condition with exponents ${\lambda _{1}}$ and ${\lambda _{2}}<{\lambda _{1}}$ also has a modification that is Hölder up to order ${\lambda _{2}}$.
Theorem 6.
(i) The process $\{{X_{t}},\hspace{0.2778em}t\in [{t_{0}},T]\}$ has a continuous modification that satisfies the Hölder condition up to order $\min \big(\gamma +\frac{3}{2},\hspace{0.2778em}1\big)$.
(ii) The process $\{{X_{t}},\hspace{0.2778em}t\in [0,T]\}$ has a continuous modification that is Hölder up to order $\min \big(\alpha +\beta +\gamma +\frac{3}{2},\hspace{0.2778em}\gamma +\frac{3}{2},\hspace{0.2778em}1\big)$.
Remark 3.
According to Proposition 2, the process X defined by (4) and (2) does not admit the bound $\operatorname{var}({X_{{t_{2}}}}-{X_{{t_{1}}}})<C\hspace{0.1667em}|{t_{2}}-{t_{1}}{|^{2\lambda }}$ for any $\lambda >\min \big(\gamma +\frac{3}{2},\hspace{0.2778em}1\big)$. Hence, according to [2, Theorem 1], the process X cannot be Hölder of order greater than $\min \big(\gamma +\frac{3}{2},\hspace{0.2778em}1\big)$.
The process X is self-similar with exponent $H=\alpha +\beta +\gamma +\frac{3}{2}$, whence $\operatorname{var}({X_{t}}-{X_{0}})=C\hspace{0.1667em}{t^{2H}}$ for some constant $C>0$. Hence, according to [2, Theorem 1], the process X cannot satisfy the Hölder condition of order greater than H on the interval $[0,T]$.
Thus, the process X cannot satisfy the Hölder condition of order greater than specified in Theorem 6.
Lemma 3.
Remark 4.
Note that in Lemma 3 we formulated the Hölder condition at a single point. The exponent in the Hölder condition at a single point may exceed 1, while the exponent in the Hölder condition on an interval does not exceed 1 unless the function or process is constant at that interval.
Proof of Lemma 3.
The process X is mean-square continuous, and ${X_{0}}=0$ almost surely. The variance of X is a power function: $\operatorname{var}({X_{t}})=\operatorname{var}({X_{t}}-{X_{0}})=C{t^{2H}}$ for some $C\ge 0$. Let us take the constants ${\lambda _{0}}$ and ${C_{0}}$ from (30). Since $C{t^{2H}}=\operatorname{var}({X_{t}}-{X_{0}})\le {C_{0}}{t^{2{\lambda _{0}}}}$ for all $t\in [0,T]$, the exponents H and ${\lambda _{0}}$ satisfy the inequality $0<{\lambda _{0}}\le H$. (Moreover, with view of Remark 2, $0<{\lambda _{0}}\le \min (1,H)$.)
Consider the stochastic process $Y=\{{Y_{s}},\hspace{0.2778em}s\in [0,{T^{H/{\lambda _{0}}}}]\}$ with ${Y_{s}}={X_{{s^{{\lambda _{0}}/H}}}}$. For all ${s_{1}}$ and ${s_{2}}$ such that $0\le {s_{1}}<{s_{2}}\le {T^{H/{\lambda _{0}}}}$, the incremental variances of Y are
\[\begin{aligned}{}& \operatorname{var}({Y_{{s_{2}}}}-{Y_{{s_{1}}}})=\operatorname{var}({X_{{s_{2}^{{\lambda _{0}}/H}}}}-{X_{{s_{1}^{{\lambda _{0}}/H}}}})\\ {} & \hspace{1em}={\left(\frac{{s_{2}^{{\lambda _{0}}/H}}}{T}\right)^{\hspace{-0.1667em}2H}}\operatorname{var}\big({X_{T}}-{X_{T{s_{1}^{{\lambda _{0}}/H}}{s_{2}^{-{\lambda _{0}}/H}}}}\big)\le \frac{{C_{0}}{s_{2}^{2{\lambda _{0}}}}}{{T^{2H}}}{\left(T-\frac{T{s_{1}^{{\lambda _{0}}/H}}}{{s_{2}^{{\lambda _{0}}/H}}}\right)^{\hspace{-0.1667em}2{\lambda _{0}}}}.\end{aligned}\]
With $0\le \frac{{s_{1}}}{{s_{2}}}<1$, the inequality $0<\frac{{\lambda _{0}}}{H}\le 1$ implies ${\bigg(\frac{{s_{1}}}{{s_{2}}}\bigg)^{\hspace{-0.1667em}{\lambda _{0}}/H}}\ge \frac{{s_{1}}}{{s_{2}}}$. Hence,
\[ T-\frac{T{s_{1}^{{\lambda _{0}}/H}}}{{s_{2}^{{\lambda _{0}}/H}}}\le T-\frac{T{s_{1}}}{{s_{2}}},\]
and
\[ \operatorname{var}({Y_{{s_{2}}}}-{Y_{{s_{1}}}})\le \frac{{C_{0}}{s_{2}^{2{\lambda _{0}}}}}{{T^{2H}}}{\left(T-\frac{T{s_{1}}}{{s_{2}}}\right)^{\hspace{-0.1667em}2{\lambda _{0}}}}=\frac{{C_{0}}\hspace{0.1667em}{({s_{2}}-{s_{1}})^{2{\lambda _{0}}}}}{{T^{2H-2{\lambda _{0}}}}}.\]
Therefore Y has a modification $\widetilde{Y}$ whose paths are Hölder up to order ${\lambda _{0}}$:
\[\begin{array}{l}\displaystyle \forall \theta \in (0,{\lambda _{0}})\hspace{0.2778em}\exists {C_{2}}={C_{2}}(\theta ,\omega )<\infty \hspace{0.2778em}\text{a.s.},\hspace{0.2778em}\forall {s_{1}},{s_{2}}\in [0,{T^{H/{\lambda _{0}}}}]:\\ {} \displaystyle |{\widetilde{Y}_{{s_{2}}}}-{\widetilde{Y}_{{s_{1}}}}|\le {C_{2}}\hspace{0.1667em}|{s_{2}}-{s_{1}}{|^{\theta }}.\end{array}\]
The process $\widetilde{X}=\{{\widetilde{X}_{t}},\hspace{0.2778em}t\in [0,T]\}$ with ${\widetilde{X}_{t}}={\widetilde{Y}_{{t^{H/{\lambda _{0}}}}}}$ is a modification of the process X. Then
whence
Substituting $s={t^{H/{\lambda _{0}}}}$ and $\theta =\lambda {\lambda _{0}}/H$ for $\lambda \in (0,H)$, we obtain (31). □
The next result is an immediate consequence of Lemma 3. Self-similarity of X is established in Proposition 1.
Theorem 7.
Let the process X be defined by (4) and (2). Then X has a modification whose paths satisfy the Hölder condition up to order $\alpha +\beta +\gamma +\frac{3}{2}$ at point $0:$
\[ \forall \lambda \in \big(0,\hspace{0.1667em}\alpha +\beta +\gamma +\frac{3}{2}\big)\hspace{0.2778em}\exists C=C(\lambda ,\omega )<\infty \hspace{0.2778em}\textit{a.s.}\hspace{0.2778em}\forall t\in [0,T]\hspace{0.1667em}:\hspace{0.1667em}|{\widetilde{X}_{t}}-{\widetilde{X}_{0}}|\le C\hspace{0.1667em}{t^{\lambda }},\]
where C is an a.s. finite random variable.