This paper makes a significant contribution by focusing on estimating the coefficients of a sample of non-linear time series, a subject well-established in the statistical literature, using bilinear time series. Specifically, this study delves into a subset of bilinear models where Generalized Autoregressive Conditional Heteroscedastic (GARCH) models serve as the white noise component. The methodology involves applying the Klimko–Nilsen theorem, which plays a crucial role in extracting the asymptotic behavior of the estimators. In this context, the Generalized Autoregressive Conditional Heteroscedastic model of order (1,1) noted that the GARCH (1,1) model is defined as the white noise for the coefficients of the example models. Notably, this GARCH model satisfies the condition of having time-varying coefficients. This study meticulously outlines the essential stationarity conditions required for these models. The estimation of coefficients is accomplished by applying the least squares method. One of the key contributions lies in utilizing the fundamental theorem of Klimko and Nilsen, to prove the asymptotic behavior of the estimators, particularly how they vary with changes in the sample size. This paper illuminates the impact of estimators and their approximations based on varying sample sizes. Extending our study to include the estimation of bilinear models alongside GARCH and GARCH symmetric coefficients adds depth to our analysis and provides valuable insights into modeling financial time series data. Furthermore, this study sheds light on the influence of the GARCH white noise trace on the estimation of model coefficients. The results establish a clear connection between the model characteristics and the nature of the white noise, contributing to a more profound understanding of the relationship between these elements.

In this article, we derive various lower bounds for the numerical radius of operators that refine the well-known inequality \(w^{2}(A)\geq \frac{1}{4} \left\Vert A^{\ast }A+AA^{\ast }\right\Vert \).

This paper introduces a novel numerical approach for solving fractional stochastic differential equations (FSDEs) using bilinear time-series models, driven by the Caputo–Katugampola (C-K) fractional derivative. The C-K operator generalizes classical fractional derivatives by incorporating an additional parameter, enabling the enhanced modeling of memory effects and hereditary properties in stochastic systems. The primary contribution of this work is the development of an efficient numerical framework that combines bilinear time-series discretization with the C-K derivative to approximate solutions for FSDEs, which are otherwise analytically intractable due to their nonlinear and memory-dependent nature. We rigorously analyze the impact of fractional-order dynamics on system behavior. The bilinear time-series framework provides a computationally efficient alternative to traditional methods, leveraging multiplicative interactions between past observations and stochastic innovations to model complex dependencies. A key advantage of our approach is its flexibility in handling both stochasticity and fractional-order effects, making it suitable for applications in a famous nuclear physics model. To validate the method, we conduct a comparative analysis between exact solutions and numerical approximations, evaluating convergence properties under varying fractional orders and discretization steps. Our results demonstrate robust convergence, with simulations highlighting the superior accuracy of the C-K operator over classical fractional derivatives in preserving system dynamics. Additionally, we provide theoretical insights into the stability and error bounds of the discretization scheme. Using the changes in the number of simulations and the operator parameters of Caputo–Katugampola, we can extract some properties of the stochastic fractional differential model, and also note the influence of Brownian motion and its formulation on the model, the main idea posed in our contribution based on constructing the fractional solution of a proposed fractional model using known bilinear time series illustrated by application in nuclear physics models.