On the Fractional Riemann-Liouville Integral of Gauss-Markov processes and applications

Abstract

We investigate the stochastic processes obtained as the fractional Riemann-Liouville integral of order α ∈ (0, 1) of Gauss-Markov processes. The general expressions of the mean, variance and covariance functions are given. Due to the central rule, for the fractional integral of standard Brownian motion and of the non-stationary/stationary Ornstein-Uhlenbeck processes, the covariance functions are carried out in closed-form. In order to clarify how the fractional order parameter α affects these functions, their numerical evaluations are shown and compared also with those of the corresponding processes obtained by ordinary Riemann integral. The results are useful for fractional neuronal models with long range memory dynamics and involving correlated input processes. The simulation of these fractional integrated processes can be performed starting from the obtained covariance functions. A suitable neuronal model is proposed. Graphical comparisons are provided and discussed.

On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes

“…Here, we also show how varying the Hurst index H of the FBM affects the integral of the considered process, and if this can be possibly put into relation with the fractional RL integral of a GM process, already studied in [15].…”

“…that we call the integrated FBM (IFBM). In particular, we study its statistical property; this is analogous to what done in [15], where we have studied the Riemann-Liouville integral of order α ∈ (0, 1) of a GM process Y(t), which is…”

“…(2α+1)Γ 2 (α+1) , α ∈ (0, 1) (see [15]). In fact, for 0 < H < 1/2, taking α = H + 1/2, we get var(I H (t)) = Γ 2 (H + 3/2)var(I H+1/2 (B)(t)); in general, for H ∈ (0, 1), we have var(I H (t)) = (2H+1)Γ 2 (H+1)t…”

“…Similarly to what done for integrated GM processes in [11], we then considered the fractional Riemann-Liouville (RL) integral of order α ∈ (0, 1) of a GM process. In [15], specifically, we studied the qualitative behavior of such integrated processes, as varying the parameter α, that turned out to have a key rule in the determination of how much memory has to be included in the stochastic description.…”

“…Note that, while in [15] the fractional RL integral of a GM process was studied, here we deal with its fractional counterpart concerning the noise: precisely, we consider the ordinary Riemann integral, but involving pseudo-fractional GM processes instead of the classical GM ones. In other words, the fractional feature is shifted from the integral operator to the stochastic term that, in the models, is involved as the noise process.…”