New Optional Stopping Theorems and Maximal Inequalities on Stochastic   Processes
  In this paper, we develop new optional stopping theorems for scenarios where the stopping rules are defined by bounded continuity regions. Moreover, we establish a wide variety of inequalities on the supremums and infimums of functions of stochastic processes over the whole range of time indexes. 
Xinjia Chen
math.PR math.ST stat.TH