Ruin Theory (Collective Risk Model)

A freelance salesman is paid $R$ units for each sale, and commissions arrive at the instants of a Poisson process of rate $\lambda$. Living costs consume his resources at unit rate. If his initial wealth is $S > 0$, show that the probability that he ever becomes bankrupt is $a^{S / R}$, where $a$ is the smallest $x > 0$ such that $x = e^{\lambda R(x-1)}$.

Prove that $a \in (0,1)$ if $\lambda R > 1$, while $a = 1$ if $\lambda R < 1$.

An insurance company receives premiums (net of costs) at unit rate per unit time, and the claims of size $X_{1}, X_{2}, \ldots$ are independent random variables with common distribution function $F$, arriving at the instants of a Poisson process of rate 1. Show that the probability $r(y)$ of ruin (that is, assets becoming negative) for an initial capital $y$ satisfies

Draw a typical trajectory of the process $Y_{t}$ in the Sparre Andersen insurance model.