Abstract
•The paper focuses on the effect of Gaussian noise on a class of fractal-fractional differential equation in the Caputo sense with power law type kernel.•The existence and uniqueness result was proven using Banach’s fixed point theorem/ contraction principle.•The upper growth moment bound was also estimated using Ito Isometry and we obtained a precise exponential growth.
We investigate a class of time-fractal-fractional Stochastic differential equation with the Atangana’s fractal-fractional differential operator in Caputo sense with power law type kernel. The upper growth bound of the random solution to the equation is estimated, and the result shows that the second moment of the solution grows exponentially at most at a precise rate. The existence and uniqueness result of the solution is also established via Banach fixed point theorem and contraction principle. We also show that the solution exhibits some long time asymptotic behaviours and some form of mean square exponential stability property.