CS30: Gaussian Processes for Fractional Dynamics and Limiting Behaviour

Organizer: Barbara Martinucci & Enrica Pirozzi (Salerno & Campania Luigi Vanvitelli)

On some fractional stochastic models based on Mittag-Leffler integrals

Enrica Pirozzi

Starting from the Prabhakar integral \[\label{Prab} \int _0 ^t (t-s)^{ \beta -1} E_{\alpha,\beta}^\gamma((t-s)^\beta \lambda) f(s) ds ,\quad \forall t\in \mathbb R^+,\] where \(E_{\alpha,\beta}^\gamma(z)\) is the generalized three-parameters Mittag-Leffler function: \[\label{Prabf} E_{\alpha,\beta}^\gamma(z)=\sum_{k=0}^{\infty}\frac{\Gamma(\gamma+k)z^k}{k!\Gamma(\gamma)\Gamma(\alpha k+\beta)},\] we consider the special case of Mittag-Leffler (ML) fractional integrals \[\int_0^t(t-s)^{\nu-1}E_{\nu,\nu}\left((t-s)^\nu \lambda\right)f(s)ds\] with \(\nu \in (0,1),\) and in particular the ML fractional stochastic integrals \[\int_0^t(t-s)^{\nu-1}E_{\nu,\nu}\left((t-s)^\nu \lambda\right)f(s)dW_s,\] with \(W\) Brownian motion. We construct models based on solution processes of fractional stochastic differential equations involving the above integrals. By introducing the ML integral operators we study such processes. Under specified assumptions, these processes are Gaussian and correlated. We are able to provide the explicit form of their covariance. We show how the analysis of the covariance turns out useful to investigate properties and refinements of the corresponding models.

Bibliography
\([1]\) Pirozzi E., Mittag–Leffler Fractional Stochastic Integrals and Processes with Applications. Mathematics, 12, 3094, 2024
\([2]\) Pirozzi, E. Some Fractional Stochastic Models for Neuronal Activity with Different Time-Scales and Correlated Inputs. Fractal and Fractional, 8(1):57, 2024
\([3]\) Leonenko N., Pirozzi E., The time-changed stochastic approach and fractionally integrated pro- cesses to model the actin-myosin interaction and dwell times. Mathematical Biosciences and Engineering 2025, Volume 22, Issue 4: 1019-1054. doi: 10.3934/mbe.2025037

Fractional rough diffusion Bessel processes: reflection, asymptotic behavior and parameter estimation

Yuliya Mishura

This is a common talk with K. Ralchenko and A. Yurchenko-Tytarenko. We consider fractional stochastic differential equations that recreate Cox-Ingersoll-Ross, Ornstein-Uhlenbeck and Bessel dynamics. The form of the equations can be different for big (\(H\in(1/2,1)\)) and small (\(H\in(0,1/2)\)) values of Hurst index \(H\). In the rough case reflection functions for CIR and Bessel dynamics participate. We establish the asymptotic in time behaviour of the solution and the reflection. Statistical parameter estimation is provided. The results are presented in \([1-4]\).

Bibliography

\([1]\) Mishura, Y., Ralchenko, K. Fractional diffusion Bessel processes with Hurst index \(H\in(0, 1/2)\). Statistics and Probability Letters, 206, 110008, 2024, pp.1–8.

\([2]\) Mishura, Y., Yurchenko-Tytarenko, A. Parameter estimation in rough Bessel model. Fractal and Fractional, 7(7), 508, 2023, pp. 1–17. Jane Smith. "Title of the Article." Journal Name, vol. X, no. Y, Year, pp. Z.

\([3]\) Mishura, Y., Yurchenko-Tytarenko, A. Standard and fractional reflected Ornstein–Uhlenbeck processes as the limits of square roots of Cox–Ingersoll–Ross processes. Stochastics, 95(1), 2023, pp. 99-117.

\([4]\) Ascione, G., Mishura, Y., Pirozzi, E. (2023). Fractional deterministic and stochastic calculus (Vol. 4). Walter de Gruyter GmbH & CoKG.

Finite-velocity random motions governed by a modified Euler-Poisson-Darboux equation

Barbara Martinucci

We study a modified Euler-Poisson-Darboux equation, whose solution is the probability density function of a non-homogeneous telegraph process \((X_t)_{t\geq 0}\). The latter starts its motion with velocity \(c>0\) and is characterized by velocity changes governed by a non-homogeneous Poisson process with intensity function \(\lambda(t)=\frac{\alpha}{\sqrt{t}}\), \(\alpha>0\), \(t>0\). We determine the Cauchy problem for the characteristic function of \(X_t\) and solve it by making use of a suitable transformation which leads to a one-dimensional Schr\(\ddot{o}\)dinger-type differential equation. The solution of such equation, which, to the best of our knowledge, is unknown in the literature, is obtained by means of the Frobenius method and provides a closed form expression of the characteristic function of \(X_t\). We also disclose the probability law of \(X_t\) and study its main features. Finally, under Kac’s-type scaling conditions, it is shown that the characteristic function of \(X_t\) tends to that of a scaled self-similar centered Gaussian process with variance proportional to \(t^{3/2}\).

Bibliography

\([1]\) S.K. Foong, U. van Kolck. Poisson Random Walk for Solving Wave Equations. Prog. Theor. Phys. 87 (2) (1992) 285–292.

\([2]\) B. Martinucci, S. Spina. On a finite-velocity random motion governed by a modified Euler-Poisson-Darboux equation. Submitted.