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I am interested in constructing a specific use of the compound Poisson process. Specifically, in insurance.

Take a standard Poisson process, $N(t)$, with distribution $N(t) \sim \mathrm{Pois}(\lambda t)$.

Take $X(t)$ to be a compound Poisson process with $Y_{i}$ jumps at random times and Poisson distributed with rate $\lambda$ per hour.

Suppose $X(t)$ models the arrival of claims at an insurance office, which arrive at a rate $c$ per hour. Also, suppose we have an initial sum $\mu$.

Find the probability that the company goes bankrupt and find the value of time at which the company goes bankrupt.

We have the survival to time $t$ to be $S(t) = \mu + ct - X(T)$.

$X(t) = \sum_{i=1}^{N(t)} Y_i$.

We have the properties $\mathbb{E}[X(t)]=\mu \lambda t$ and $\mathbb{V}[X(t)] = (\sigma ^2 + \mu ^2) \lambda t $.

Suppose we have $Y_{i} \sim N(2,9)$ and rates arrive at $c=1$ per hour, with initial sum $\mu = 20$ and rate $\lambda = 1$.

Then $\mathbb{E}[X(t)] = 2 \lambda t = 2t$ and $\mathbb{V}[X(t)] = 13t$.

So we have want to find the probability of $S(t) = 0$ and the time at which this occurs.

$P(S(t)=0) = P(20+2t=X(t)) = P(\sum_{i=1}^{N(t)} Y_i = 20 +2t) $

?? At around this point I lose sense of the question. Can someone put some positive direction into it? Perhaps it is formulated incorrectly (or inappropriately!).

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