# Help with Durrett 5.7.8

I've been working on Durrett Exercise 5.7.8 without any avail: Let $S_n$ be the total assets of an insurance company at the end of year $n$. In year $n$, premiums totaling $c>0$ are received and claims $\zeta_n\sim N(\mu,\sigma^2)$ are paid where $\mu<c$. To be precise, if $\xi_n= c - \zeta_n$ then $S_n = S_{n-1} + \xi_n$. The company is ruined if its assets drop to 0 or less. Show that if $S_0 > 0$ is nonrandom, then

$P($ruin$)\leq \exp(-2(c-\mu)S_0/\sigma^2)$.

The way I've thought about this problem is that if $\tau$ is the time of bankruptcy, we are looking for $P(\tau < \infty)$. Obviously, Wald's Identity isn't very useful.

If I define $X_n = S_n - n(c-\mu)$, then $X_n$ is a martingale. However, I can't find any theorems regarding the stopping times of martingales or the probability of them being infinite. Durrett's theorems mostly involve expectations of $\tau$, which in our case are clearly infinite.

• To make $e^{-\tau S_n}$ a martingale, find a positive root of $E(e^{-\tau \xi})=1$. Since $S_n\to\infty$, conclude that $e^{-\tau S_0}=P(ruin)E(e^{-\tau R})$ where $R\le 0$ are your assets upon ruin. From here you get $P(ruin)\le e^{-\tau S_0}$. If you knew that $\xi\ge -A$ (not the case here) then you could also an easy lower bound for $P(ruin)$. – A.S. Feb 24 '16 at 4:23

Note that $$\mathbb E\left[e^{-\left(\frac{2(c-\mu)}{\sigma^2}\right)\xi_1}\right] = e^{-\left(\frac{2(c-\mu)}{\sigma^2}\right)(c-\mu) -\frac12 \left(\frac{2(c-\mu)}{\sigma^2}\right)^2\sigma^2}=e^0=1.$$ Set $\theta=\frac{-2(c-\mu)}{\sigma^2}$ and $X_n=S_0-S_n$, then $\{e^{-\theta X_n}\}$ is a martingale. Let $B>S_0$ and define $$N=\inf\{n>0: X_n>S_0\text{ or } X_n<-B\}.$$ Then by optional stopping, $$\mathbb E\left[e^{-\theta X_N}\right] = 1,$$ and so $$1 = \mathbb E\left[e^{-\theta X_N}\mid X_N>S_0\right]\mathbb P(X_N>S_0) + \mathbb E\left[e^{-\theta X_N}\mid X_N<-B\right]\mathbb P(X_N<-B).$$ Since $-\theta>0$, we have $$1\geqslant e^{-\theta S_0}\mathbb P(X_N>S_0),$$ and letting $B\to\infty$ yields $$\mathbb P(\mathrm{ruin})\leqslant e^{-\left(\frac{2(c-\mu)}{\sigma^2}\right)S_0}.$$