Hope that my English is readable after all... In the textbook it is written:
Consider the process $U_t=u+ct-\sum\limits_{i=1}^{N(t)}y_i$ where $y_i\geq 0\,$ iid, $(N(t))_{t\geq 0}$ is a Poisson proces with intensity $\lambda$ independent from $(y_i)_{i\geq 1}$. Define $\phi(u)=\mathsf P(\exists t> 0, U_t< 0)$.
- If $E(y_i)=\mu, c> \lambda \mu \,$, then $ \phi(0)=\frac{\lambda \mu}{c}\,; $
- Let $F(x)$ be the distribution of $y_1$, and $r_0> 0$ which satisfied: $ \int_0^\infty e^{r_0x}(1-F(x))dx=\frac{c}{\lambda}\,, $ then$\phi(u)\leq e^{r_0x}$, where $(u\geq 0)$
I feel really confused about the second conclusion, could anybody help to explain how to proof them? Thanks.