International Monthly Seminar on Time Scales Analysis #8

In this work, we investigate the approximation of spatially nonstationary spatio-temporal GARCH (ST-GARCH) processes by spatially stationary counterparts at fixed locations. This approach enables a localized analysis of complex spatio-temporal volatility structures. Building upon the model's recursive formulation, we establish that the ST-GARCH process can be represented as a sum of random matrix products, allowing us to derive conditions under which the process admits a Lipschitz continuous approximation. We prove that, under mild regularity and continuity assumptions, the nonstationary process (X^2_t(s)) can be closely approximated by a spatially stationary process (X^2_{t,s_0}(s)) at a fixed point (s_0), with a convergence rate governed by the spatial distance (|s - s_0|_\infty). Furthermore, using a Taylor expansion and a derivative-based construction, we refine this approximation by including the first-order spatial derivative, yielding an improved representation as a linear combination of two spatially stationary processes. Our theoretical findings lay the groundwork for practical localized modeling and inference in real-world applications involving heterogeneous spatio-temporal data.

[1]Naimi Abdellouahab [2]Messaouda Benattia
This presentation give a numerical framework for solving a class of Caputo fractional integro-di erential problem with integral condition. In a previous work [3], the existence and uniqueness of solutions were established under suitable Lipschitz conditions. In the present paper, a specific example is constructed to validate these theoretical results numerically, and an Artificial Neural Network (ANN) method is applied to approximate the solution. The ANN approach incorporates the fractional operators into the loss function using numerical quadrature techniques. The numerical results demonstrate high accuracy and good convergence behavior, confirming the e ectiveness of the proposed numerical scheme.