Dynamical behavior of tempered φ-Caputo type fractional order stochastic differential equations driven by Lévy noise

This paper focuses on the analysis of a class of stochastic differential equations with tempered $\phi$-Caputo fractional derivative ($\phi$-CFD) and Lévy noise. We propose comprehensive mathematical techniques to address the existence, uniqueness and stability of solution to this equation. For existence and uniqueness, the Picard–Lindelof successive approximation technique is used analyze the results. Also, We use Mittag-Leffler (M-L) function to investigate the stability of the solution. This research applies the broad understanding of stochastic processes and fractional differential equations, as well as known results, to the analysis of systems with tempered $\phi$-CFD. These equations capture complex phenomena in the field of financial assets, making their investigation on the stock prices particularly valuable.