Existence and uniqueness results for double jump fractional uncertain differential equations

Abstract

Fractional uncertain differential equations have been used to model random processes in economics and other fields that exhibit jumps, dependency, and nonlinearities, and which possess uncertainties due to limited data and inadequate models. In this paper, a double V-jump fractional uncertain differential equation (DV-FUDE) is presented as a $\theta^{th}$-order Riemann-Liouville or Caputo fractional uncertain differential equation with the addition of two V-jump independent processes on different filtrations. The equation models systems possessing two sources of uncertain shocks attributed to internal and external factors, respectively. Exact solutions in the case of time-dependent coefficients are given in terms of the Mittag-Leffler function. Sample continuity, existence, and uniqueness for the general Riemann-Liouville and Caputo DV-FUDE are established using the Banach Fixed Point Theorem, under global Lipschitz and linear growth conditions on the coefficients. Some extensions and possible areas of application are highlighted for future research.