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 Nov 8 comment Breaking up Wiener processes with indicator functions? Okay, so can I confirm that this is the correct argument: (i)$W_t(\omega) = {1}_{\{W_t(\omega) \geq 0\}}W_t(\omega) + {1}_{\{W_t(\omega) < 0\}}W_t(\omega)$ holds for each $\omega \in \Omega$. (ii)Therefore, $W_t = {1}_{\{W_t \geq 0\}}W_t + {1}_{\{W_t < 0\}}W_t$ is true (almost surely). Nov 8 comment Breaking up Wiener processes with indicator functions? Okay, so you're suggesting that I use the fact that the identities I've stated hold path-wise (i.e. $\text{P-a.s.}$ under filtration equipped probability space with measure $P$), and therefore hold in all possible states of the world, and therefore the expressions are correct as they're stated. Nov 8 asked Demonstrate that every martingale is a local martingale. Nov 8 awarded Commentator Nov 8 comment Breaking up Wiener processes with indicator functions? So this confirms the two identifies path-wise. (i.e. for $W_t(\omega)$ and $|W_t(\omega)|$). However, I'm not quite sure how this can then be extended to demonstrate the case for the full $W_t$ random variable? Nov 8 comment Sum of two stopping times is a stopping time? @did Thanks for your help mate! Nov 8 comment Min of two stopping times is also a stopping time. So you're saying that in fact $\{min(\sigma,\tau)\leq t\} =\{\omega_1 : \sigma(\omega_1) \leq t\} \cup \{\omega_2 : \tau(\omega_2) \leq t\}$ Nov 8 accepted Min of two stopping times is also a stopping time. Nov 8 revised Breaking up Wiener processes with indicator functions? deleted 63 characters in body Nov 8 comment Min of two stopping times is also a stopping time. You're right! Thanks. Nov 8 asked Min of two stopping times is also a stopping time. Nov 8 asked Breaking up Wiener processes with indicator functions? Nov 8 revised Sum of two stopping times is a stopping time? edited title Nov 8 comment Sum of two stopping times is a stopping time? @StefanHansen Okay then I'm lost in this problem unfortunately. Nov 8 comment Sum of two stopping times is a stopping time? I thought that that is true because of the following: Since I'm told that $\sigma$ is a stopping time, then $\{\sigma \leq x\} \in \mathscr{F}_x$ for any $x \in [0,\infty)$. Now for $x = t - \tau$, it's true that the image $x(\omega)$ satisfies the above, but I'm not sure if $x$ itself does. I'm pretty bad at maths (unfortunately). Nov 8 asked Sum of two stopping times is a stopping time? Oct 17 accepted Continuity and pushing a limit inside the function's domain Oct 17 comment Continuity and pushing a limit inside the function's domain Yes, it does, Thank YOU! Oct 17 comment Continuity and pushing a limit inside the function's domain (+1) I have a related question. Is this proposition true or false: $(f\text{ is Right continuous})\Rightarrow \neg (f\text{ is Left continuous})$ Oct 17 revised Continuity and pushing a limit inside the function's domain edited body