Let $W$ be a standard Brownian motion under given probability space. For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time $T^a = \inf(t>0:W(t) = a)$. That is, $W^a(t) = W(t ) 1_{(t\le T^a)} + W(T^a) 1_{(t> T^a)}$\wedge T^a)$.
We want to consider the following question: Is the process $W^1$ of the class DL?

(Solution1): Yes. Indeed, for any fixed $t>0$, we can prove the collection of random variables $( W(s), 0< s< t)$ is uniformly integrable by definition, since $E [|W^1(t)|] < \infty$.

We provide completely different answer using the following proposition from the Problem 1.5.19 (i) of Book [Karazas and Shereve 98].

[Proposition] A local martingale of class DL is martingale.

(Solution2): No. $W^1$ is strict local martingale, since $E [W^1(T^1)] = 1> E [W(0)]$. By [Proposition], $W^1$ is not of class DL.

In the above, we obtained completely two different solutions. Where is wrong?

Let $W$ be a standard Brownian motion under given probability space. For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time $T^a = \inf{t>0: W(t) inf(t>0:W(t) = a}$a)$. That is, $W^a(t) = W(t) 1_{{t\le T^a}1_{(t\le T^a)} + W(T^a) 1_{{t> T^a}}$1_{(t> T^a)}$.
We want to consider the following question: Is the process $W^1$ of the class DL?

[Solution1]:

(Solution1): Yes. Indeed, for any fixed $t>0$, we can prove the collection of random variables ${W(s):0 ( W(s), 0< s< t)$ is uniformly integrable by definition, since $E [|W^1(t)|] < \infty$.

We provide completely different answer using the following proposition from the Problem 1.5.19 (i) of Book [Karazas and Shereve 98].

[Proposition] A local martingale of class DL is martingale.

[Solution2]:

(Solution2): No. $W^1$ is strict local martingale, since $E [W^1(T^1)] = 1> E [W(0)]$. By [Proposition], $W^1$ is not of class DL.

In the above, we obtained completely two different solutions. Which one Where is not correctwrong?

Let $W$ be a standard Brownian motion under given probability space. For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time $T^a = \inf{t>0: W(t) = a}$. That is, $W^a(t) = W(t) 1_{{t\le T^a}} + W(T^a) 1_{{t> T^a}}$.
We want to consider the following question: Is the process $W^1$ of the class DL?

[Solution1]: Yes. Indeed, for any fixed $t>0$, we can prove ${W(s):0

We provide completely different answer using the following proposition from the Problem 1.5.19 (i) of Book [Karazas and Shereve 98].

[Proposition] A local martingale of class DL is martingale.

[Solution2]: No. $W^1$ is strict local martingale, since $E [W^1(T^1)] = 1> E [W(0)]$. By [Proposition], $W^1$ is not of class DL.

In the above, we obtained completely two different solutions. Which one is not correct?