Cramer-Lundberg risk model

What do the sample paths of process (11) look like?

Example 3. On some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ consider two sequences of nonnegative random variables $\left\{ \xi_{j}, j \in \mathbb {N}\right\}$ and $\left\{ \eta_{j}, j \in \mathbb {N}\right\}$ such that $\left\{ \xi_{j}, \eta_{j}, j \in \mathbb {N}\right\}$ are jointly independent and $\operatorname {Law}\left(\xi_{j}\right) = \operatorname {Law}\left(\xi_{1}\right)$, $\operatorname {Law}\left(\eta_{j}\right) = \operatorname {Law}\left(\eta_{1}\right)$, $j \in \mathbb {N}$. For positive constants $y_{0}$ and $c$ define the process

where $X_{t}(\omega )$ is given by formula (10).

Example 2. For a sequence $\xi_{1}, \xi_{2}, \ldots$ of independent identically distributed nondegenerate nonnegative random variables, the renewal process is defined by the formula

Throughout, the sum over the empty set is taken to be zero, so $X_{t}(\omega ) = 0$ if $\xi_{1}(\omega ) > t$.

Let $r(y)$ be the ruin function of the insurance problem (8.10.7). In the notation of that problem, show that the Laplace-Stieltjes transforms $r^{*}(\theta )$ and $F^{*}(\theta )$ are related by $\lambda F^{*} = r^{*}\left(\theta -\lambda +\lambda F^{*}\right)$.

An insurance company pays independent and identically distributed claims $\left\{ K_{n}: n \geq 1\right\}$ at the instants of a Poisson process with intensity $\lambda$, where $\lambda \mathbb {E}\left[K_{1}\right] < 1$. Premiums are received at constant rate 1. Show that the maximum deficit $M$ the company will ever accumulate has moment generating function

Let $£ Y_{n}$ be the assets of an insurance company after $n$ years of trading. During each year it receives a total (fixed) income of $£ P$ in premiums. During the $n$th year it pays out a total of $£ C_{n}$ in claims. Thus $Y_{n+1} = Y_{n}+P-C_{n+1}$. Suppose that $C_{1}, C_{2}, \ldots$ are independent $N\left(\mu , \sigma^{2}\right)$ variables and show that the probability of ultimate bankruptcy satisfies

An insurance company receives premiums (net of costs) at rate $\rho$ per unit time. Claims $X_{1}, X_{2}, \ldots$ are independent random variables with the exponential distribution with parameter $\mu$, and they arrive at the times of a Poisson process of rate $\lambda$ (assume the usual independence, as well as $\lambda , \mu , \rho > 0)$. Let $Y(t)$ be the assets of the company at time $t$ where $Y(0) = y > 0$. Show that

where $\theta = \mu -(\lambda / \rho )$.