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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 $\phi(u)=\mathsf P(\exists t> 0, U_t< 0)$.

  1. If $E(y_i)=\mu, c> \lambda \mu \,$, then $ \phi(0)=\frac{\lambda \mu}{c}\,; $
  2. 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.

  • 0
    Cannot write full answer now, but two useful books are [Non-life Insurance Mathematics](http://books.google.com/books/about/Non_Life_Insurance_Mathematics.html?id=yOgoraGQhGEC) and [Ruin probabilities](http://books.google.com/books?id=LblaB4XJg9wC&printsec=frontcover&dq=ruin+probabilities+asmussen&hl=en&ei=IJzHTuHcLpGe-Qb1kb$A$F&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC4Q6AEwAA#v=onepage&q=ruin%20probabilities%20asmussen&f=false)2011-11-19

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