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First $W_^1$ W^1$will be of class DL as soon as it is a martingale by proposition 16.25. So showing that$W^1$is a martingale is sufficient to prove your claim (which follows from Proposition 16.30 for example if you know that a Brownian Motion is a martingale) Second here is why Solution 2 doesn't work By proposition 16.26 if$M_\tau$is in L^1 for every bounded stopping times (which is the case here). We have$X_t$is a martingale if$E[M_\tau]=E[M_0]$for every bounded stopping time$\tau$. This is the criteria you are trying to apply to get your contradiction. The problem with this, is that$T_1$is not bounded almost surely so you cannot apply the preceding criteria to show that$W^1$is not of class DL. I hope I didn't make any mistake Regards 1 Hi kenneth Have a look at the following document http://www.ma.utexas.edu/users/gordanz/teaching/10_Spring_M385D/lecture16.pdf (In particular to Propositions 16.24,16.25, 16.26, and 16.30) First$W_^1$will be of class DL as soon as it is a martingale by proposition 16.25. So showing that$W^1$is a martingale is sufficient to prove your claim (which follows from Proposition 16.30 for example if you know that a Brownian Motion is a martingale) Second here is why Solution 2 doesn't work By proposition 16.26 if$M_\tau$is in L^1 for every bounded stopping times (which is the case here). We have$X_t$is a martingale if$E[M_\tau]=E[M_0]$for every bounded stopping time$\tau$. This is the criteria you are trying to apply to get your contradiction. The problem with this, is that$T_1$is not bounded almost surely so you cannot apply the preceding criteria to show that$W^1\$ is not of class DL.