# Show that the assumption of right-continuity in the statement of the stopping theorem cannot be omitted

In our homework assignment, we were supposed to find an example showing that the assumption of right-continuity in the statement of the stopping theorem cannot be omitted in general (cf. http://www.math.ethz.ch/education/bachelor/lectures/fs2012/math/bmsc/bmsc_fs12_04.pdf exercise 4-2 c)).

In the hint it said: For a standard exponentially distributed random variable T, consider the process $M = (M_t)_{t \geq 0}$ given by $M_t = (T \wedge t ) + 1_{ \{ t \leq T \} }$ together with the $P$-augmentation of the filtration generated by the process $(T \wedge t)$.

Moreover, we were told that we should try to prove that $M_t = E[T | \widetilde{\mathcal{F}}_t]$, were $\widetilde{\mathcal{F}}_t$ denotes the $P$-augmentation of the sigma algebra $\mathcal{F}_t = \sigma (T \wedge s ; s \leq t)$.

Well, I know that once if proven $M_t = E[T | \widetilde{\mathcal{F}}_t]$, it follows that $M$ is a uniformly integrable $\widetilde{\mathcal{F}}_t$-martingale. Also, I can show that $T$ is a $\widetilde{\mathcal{F}}_t$-stopping time. Therefore, if the assumption of right-continuity were not necessary, the stopped process $M^T$ with $M_t^T = (T \wedge t ) + 1_{ \{ T \wedge t \leq T \} } = (T \wedge t ) + 1$ would also be a uniformly integrable martingale (by the stopping theorem). Then, the difference $N_t := M_t^T - M_t = 1_{ \{ t > T \}}$ would also be a uniformly integrable martingale. But $E[N_0] = 0$ whereas $E[N_{\infty}] = E[1] = 1$, a contradiction.

Can anybody help me prove $M_t = E[T | \widetilde{\mathcal{F}}_t]$? Or would anybody happen to know a different counterexample?

Thanks a lot!

Regards, Si

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the basic idea is $T 1_{(T<t)}$ is $F_t$ measurable, $(T > t)$ is an atom in $F_t$ and $E(T \vert T > t) = T +1$ by the memoryless property of the exponential. – mike May 2 '12 at 10:56
@mike: Thanks a lot! Unfortunately, I still don't understand: do you follow from the memoryless property that $E[T 1_{ \{ t < T \} } | \mathcal{F}_t] = (1 + T) 1_{ \{ t < T \} }$ ? (Ps: I don't have to hand this exercise in, it was due long time ago (as you can see in the link above)) – Mad Si May 2 '12 at 13:04
$T1_{(t<T)}$ is is $F_t$ measurable ( in fact it is $M_t 1_{(t<T)}$ which are 2 $F_t$ measurable functions) so $\mathbb E (T1_{(t<T)} \vert F_t) = T1_(t<T)$. $\mathbb E (T1_{(T>t)} \vert F_t)= \mathbb E(T \vert T>t) 1_{(T>t)}$, which only depends on $T>t$ being an atom in $F_t$, and then u can evaluate $\mathbb E(T \vert T>t)$ using the fact that it is exponential. Discussion of filtrations generated by stopping time in Quantitative Risk Management, McNeil , Frey,Embrechts, – mike May 2 '12 at 13:49