Degradation with random jumps is a process of stochastically continuous degradation with sporadic jumps that occur at random times and have random sizes. We use Lévy subordinators to model the evolution of this type of degradation processes. Based on corresponding Fokker-Planck equations, we derive explicit results for reliability function and lifetime moments in terms of Laplace transform, represented by Lévy measures, which can model many complex jump mechanisms. By specifying different Lévy measures to describe different jump mechanisms in degradation, our model for reliability function and lifetime moments is general and can fit many different types of degradation data sets. Stochastic models based on gamma process or compound Poisson process become special cases of our model. Numerical experiments are used to demonstrate that our general model performs well and is applicable. More importantly, it provides us a new methodology to deal with multi-degradation processes under dynamic environment.