CS20: Parameter Randomization Methods for Stochastic Processes

Organizer: Marek Arendarczyk (University of Wrocław)

Anomalous diffusive processes with random parameters. Theory and Applications.

Agnieszka Wyłomańska

Anomalous diffusion processes have been widely discussed in the literature. The process \(\{X(t)\}\) is labelled as anomalous diffusive if its second moment grows in time as a power-law function, i.e., \[\label{eq1} \mathbb{E} \left(X^2(t)\right)=2Dt^{\alpha} \,,\] where \(\alpha\in[0,2]\) is the anomalous exponent and \(D\in[0,\infty)\) is the generalized diffusion coefficient (called also diffusion coefficient). The classical example of the anomalous diffusive class is the fractional Brownian motion (FBM) \(\{B_{H}(t)~t\geq 0\}\) that is continuous centred Gaussian process defined through the integral representation \([1]\) \[\tag{1} B_{H}(t) =A_{H}\Bigg[\int_{0}^{t}(t-u)^{H-1/2}d\tilde{B}(u) +\int_{-\infty}^{0} \mspace{-4mu}\left((t-u)^{H-1/2}-(-u)^{H-1/2}\right)d\tilde{B}(u)\Bigg] \mspace{-4mu},\] where \(0<H<1\) is the Hurst exponent. The process \(\{\tilde{B}(t),~t\in \mathbb{R}\}\) is the extension of ordinary Brownian motion to the negative time axis defined as \[\tilde{B}(t)= \begin{cases} B_1(t) ~ \mbox{for}~t>0,\\ B_2(-t) ~\mbox{for}~ t\leq 0, \end{cases}\] where \(\{B_1(t),~ t \geq 0\}\) and \(\{B_2(t),~ t \geq 0\}\) are two independent ordinary Brownian motions. Let us note that the Hurst exponent \(H\) determines the anomalous diffusion behavior of FBM and \(\alpha\) in the equation \((1)\) is equal to \(2H\).

However, recent empirical findings suggest that classical anomalous diffusion models often fail to adequately capture the underlying dynamics of such phenomena. Consequently, contemporary research has proposed numerous modifications to these models in order to better represent the specific behaviors observed in real data. One such modification involves the incorporation of random parameters that govern anomalous diffusion within traditional models.

During the presentation, we introduce a novel conceptual approach to anomalous diffusion modeling, with a particular focus on FBM and its multifractional extension. In these models, the Hurst exponent \(H\) is replaced by a suitable randomly chosen variable \(\mathcal{H}\) or a stochastic process \(\{\mathcal{H}(t)\}\), respectively. We examine the fundamental probabilistic properties of the models discussed and discuss methodologies for their application to empirical data. The presentation concludes with several illustrative examples based on real-world datasets. The methodology, properties and techniques used for the analysis of the new processes have been published in \([2,3]\).

Bibliography

\([1]\) B. B. Mandelbrot and J. W. V. Ness, Fractional Brownian Motions, Fractional Noises and Applications, SIAM Review 10, 422, 1968.

\([2]\) M. Balcerek, K. Burnecki, S. Thapa, A. Wyłomańska, A. Chechkin: Fractional Brownian motion with random Hurst exponent: accelerating diffusion and persistence transitions, Chaos 32, 093114, 2022

\([3]\) M. Balcerek, S. Thapa, K. Burnecki, H. Kantz, R. Metzler, A. Wyłomańska, A. Chechkin: Multifractional Brownian motion with telegraphic, stochastically varying exponent, accepted to PRL, 2025

Lévy processes with values in the cone of non-negatively defined matrices

Krzysztof

The infinite divisibility of the exponential distribution underlies the construction of the classical gamma Lévy motion. A natural generalization of the exponential distribution to the matrix-valued setting exists; however, it is not infinitely divisible and therefore cannot be directly used to define a matrix-valued gamma motion, see \([3]\).

In existing literature, see \([1]\), the Lévy–Khintchine representation on the cone of positive semidefinite matrices has been employed to construct infinitely divisible matrix-valued analogues of gamma distributions. In this work, we propose three alternative approaches that yield distinct infinitely divisible distributions on the cone, while retaining the direct use of the natural matrix-valued exponential distribution.

The first approach introduces a novel Lévy motion with multivariate argument and thus defined on the non-negative orthant, resulting in a multivariate shape (divisibility) parameter. This construction reveals a simple relationship with the classical matrix gamma distribution, which has a scalar shape parameter.

The second and third constructions involve one-dimensional divisibility parameters and can be seen as natural matrix-valued extensions of the Bondesson shot-noise representation of the one-dimensional gamma motion, see \([2]\), where the matrix exponential distribution governs the jump sizes.

Compared to prior constructions, the proposed models offer advantages such as explicit analytical expressions for many of their properties and straightforward simulation algorithms. These features make the models well-suited for applications in stochastic modeling, especially where stochastic integration is required and both theoretical analysis and computation must remain tractable.

Bibliography

\([1]\) Pérez-Abreu, V. and Stelzer, R. (2014). “Infinitely divisible multivariate and matrix gamma distributions”. Journal of Multivariate Analysis, 130:155–175.

\([2]\) Rosiński, J. (1990). “On series representations of infinitely divisible random vectors.” Ann. Probab., 18(1):405–430.

\([3]\) Kozubowski, T.J., Mazur, S., and Podgórski, K. (2025) “Matrix variate gamma distributions with unrestricted shape parameter” accepted for publication in Journal of Multivariate Analysis,

\([4]\) Kozubowski, T.J., Mazur, S., and Podgórski, K. (2022) “Matrix variate gamma distributions and Related stochastic processes.” Working Paper 12/2022 (STATISTICS), ISSN 1403-0586 Örebro University School of Business SE-701 82 Örebro, Sweden

Multiple scaled multivariate distributions and processes

Tomasz J. Kozubowski

The normal mean-variance mixture model is a versatile tool in probability and statistics, widely used in areas such as hierarchical modeling, Bayesian inference, stochastic processes, and machine learning. We explore an extension where the mixing is linked to the eigenvalues of the Gaussian covariance matrix, leading to multiple-scaled Gaussian mixture models - recently proposed for more flexible clustering and classification (see, e.g., \([1-5]\)). We further modify this framework to define a multivariate distribution uniquely determined by the underlying Gaussian covariance matrix. Key properties such as infinite divisibility and the related multivariate Lévy processes are discussed, highlighting their relevance in stochastic modeling. This is a joint work with Amos Natido.

Bibliography

\([1]\) Forbes, F. and Wraith, D. “A new family of multivariate heavy-tailed distributions with variable marginal amounts of tail weight: Application to robust clustering." Statistics and Computing, vol. 24, 2014, pp. 971–984.

\([2]\) Franczak, B.C., Tortora, C., Browne, R.P. and McNicholas, P.D. “Unsupervised learning via mixtures of skewed distributions with hypercube contours." Pattern Recognition Letters, vol. 58, 2015, pp.  69–76.

\([3]\) Punzo, A. and Tortora, C. “Multiple scaled contaminated normal distribution and its application in clustering." Statistical Modelling, vol.  21, 2021, pp. 332–358.

\([4]\) Tortora, C., Franczak, B.C., Bagnato, L. and Punzo, A. “A Laplace-based model with flexible tail behavior." Computational Statistics and Data Analysis, vol. 192, 2024, pp. 107909.

\([5]\) Wraith, D. and Forbes, F. “Location and scale mixtures of Gaussians with flexible tail behaviour: Properties, inference and application to multivariate clustering." Computational Statistics and Data Analysis, vol. 90, 2015, pp. 61–73.