How badly are the Burholder-Davis-Gundy inequalities affected by   arbitrary random times?
  This note deals with the question: what remains of the Burkholder-Davis-Gundy inequalities when stopping times $T$ are replaced by arbitrary random times $\rho $? We prove that these inequalities still hold when $T$ is a pseudo-stopping time and never holds for ends of predictable sets. 
Ashkan Nikeghbali