A Generalization of the Submartingale Property: Maximal Inequality and Applications to Various Stochastic Processes

We generalize the notion of the submartingale property and Doob's inequality. Furthermore, we show how the latter leads to new inequalities for several stochastic processes: certain time series, Levy processes, random walks, processes with independent increments, branching processes and continuous state branching processes, branching diffusions and superdiffusions, as well as some Markov processes, including geometric Brownian motion.