Inner product spaces of integrands associated to subfractional Brownian motion

Abstract

We characterize the domain of the Wiener integral with respect to a subfractional Brownian motion ${\left\{S_H\left(t\right)\right\}}_{t\ge 0},H\in (0,1),H\ne \frac{1}{2}$. The domain is a Hilbert space which contains the class of elementary functions as a dense subset. If $0<H<\frac{1}{2},$ any element of the domain is a function and if $\frac{1}{2}<H<1,$ the domain is a space of distributions. The RKHS of $S_H$ is also determined.

Introduction

The long-range dependence property became an important aspect of stochastic models in telecommunication, turbulence, finance, etc.

Roughly speaking, the long-range dependence criterion means that the evolution after time $t$ of a system depends on the whole history up to $t$ (including $t$) in the sense that there exists a strong dependence between the observations of the process in different times such that it approaches zero slowly as the distance between observation times goes to infinity.

The best known and most used process that exhibits long-range dependence is the fractional Brownian motion (fBm for short). This process was first introduced by Kolmogorov (1940) and later studied by Mandelbrot and Van Ness (1968). The fBm is a continuous centred Gaussian process ${\left\{B_H\left(t\right)\right\}}_{t\ge 0}$, starting from zero, with covariance

$$E\left[B_H\left(t\right)B_H\left(s\right)\right]=\frac{1}{2}(s^{2H}+t^{2H}-{|t-s|}^{2H})\text{,}s,t\ge 0\text{,}$$

where $H\in (0,1)$ ($H$ is called Hurst parameter). The case $H=\frac{1}{2}$ corresponds to the Brownian motion.

The self-similarity and stationarity of the increments are two main properties for which fBm enjoyed success as a modelling tool. The fBm is the only continuous Gaussian process which is self-similar and has stationary increments.

An extension of Bm which preserves many properties of the fBm, but not the stationarity of the increments, has been proposed by Bojdecki et al. (2004a).

In Bojdecki et al. (2004a) so called subfractional Brownian motion (sub-fBm for short) is introduced as a continuous Gaussian process ${\left\{S_H\left(t\right)\right\}}_{t\ge 0}$, starting from zero, with covariance

$$E\left[S_H\left(t\right)S_H\left(s\right)\right]=s^{2H}+t^{2H}-\frac{1}{2}[{(s+t)}^{2H}+{|t-s|}^{2H}]\text{,}s,t\ge 0\text{.}$$

For $H>\frac{1}{2}$ the sub-fBm arises from occupation time fluctuations of branching particle systems (see Bojdecki et al., 2004a, Bojdecki et al., 2004b, Bojdecki et al., 2006).

The sub-fBm has properties analogous to those of fBm (self-similarity, long-range dependence, Hölder paths and it is neither a Markov processes nor a semimartingale).

Moreover, sub-fBm has nonstationary increments and the increments over nonoverlapping intervals are more weakly correlated and their covariance decays polynomially at a higher rate in comparison with fBm (for this reason in Bojdecki et al. (2004a) it is called the sub-fBm).

Some basic properties of sub-fBm are given in Bojdecki et al. (2004a), Tudor (2007).

It is well known that in the Brownian case $L^2\left(R\right)$ is the space of integrands for the Wiener integral. In the case of fBm in Pipiras and Taqqu (2000), a similar problem is considered. The corresponding families of integrands are defined in terms of fractional integrals and derivatives.

In this paper we characterize the domain of the Wiener integral with respect to a subfractional Brownian motion ${\left\{S_H\left(t\right)\right\}}_{t\ge 0},H\in (0,1),H\ne \frac{1}{2}$. The domain is a Hilbert space which contains the class of elementary functions as a dense subset. If $0<H<\frac{1}{2}$, any element of the domain is a function and if $\frac{1}{2}<H<1$, the domain is a space of distributions. In the latter case we provide relevant examples of subspaces of functions included in the domain. The case of an interval will be considered elsewhere.

The reproducing kernel Hilbert space (RKHS for short) of $S_H$ is also determined.