Process bounded at stopped times, constant between. I have a positive pure jump  process $Y_t$, which is progressive w.r.t. the filtration ($F_t$) and has jump-times $(\tau_j)$. I know that $\sup_{j\in \mathbb{N}}\mathbb{E} Y_{\tau_j}<\infty$ and $\tau_j\rightarrow \infty$ for $j\rightarrow \infty$ almost surely. 
Does it hold that $\sup_{t<\infty}\mathbb{E}Y_t$ is bounded as well? If not, does there exist a simple counter-example? 
 A: It doesn't hold.
Counterexample:
Let $S_i$ be the inter-arrivaltimes so that $\tau_n=\sum_{i=0}^{n-1}S_i,$ and define $(Y_t)$ to be the natural induced jump process.
Define $S_0=1$. For general $n\in\mathbb{N}$ we define  the $Y_{\tau_n}'s$ pairwise independent with  $P(Y_{\tau_n}=n)=1-P(Y_{\tau_n}=n^{-1})=n^{-1}$. The inter-arrival times are defined as $S_n=1$ if $Y_{\tau_n}=n$ and otherwise $S_n=2^{-n}$.
By Borel Cantelli there are infintely many jumps where $Y_{\tau_n}=n$. Notice also that $\mathbb{E}Y_{\tau_n}=1$. To show that $\sup_t\mathbb{E}Y_t=\infty$ we fix $t\in(N-2^{-M},N]$ for large $N,M\in\mathbb{N}$. Define the random variables $c_b(t),c_s(t)$ counting big and small jumps of $Y$ before $t$ ( i.e. jumps where $Y_{\tau_n}=n$ or $Y_{\tau_n}=n^{-1}$ respectively). By construction of the IRI's we must have $c_b(t)\geq N-1$ so
$$Y_t\geq (N-1)1_{c_s(t)<M}+(N-1)^{-1}1_{c_s(t)\geq M}.$$
Note now that $1_{c_s(t)< M}\rightarrow 1$ almost surely for $M\rightarrow \infty$. This proves that there exist a sequence $t_N>N-1$ such that
$$\mathbb{E}Y_{t_N}\geq 0.99(N-1).$$
