So I have the next problem:

In order to simulate the ruin of a risk process I need, of course, to simulate the risk process itself but in this case this process has some characteristics that make it particularly hard for me.

The risk process (Compound Poisson Process) is as follows:

\begin{equation} S_t=\sum_{i=0}^{N_t}X_i \end{equation}

Where $\{N_t:t \geq 0\}$ is a Poisson Process with intensity of $\lambda$ and $X_i \sim LogNormal(\mu,\sigma^2), \forall i$ where $X_i$ are independent of $N_t$.

First Idea

So, my first idea was to try to simulate this by using the Inverse Transform Technique: I would try to find the distribution of $S_t$, find its inverse function, and simulate it using values fromm a $U(0,1)$.

Following the above mentioned path I would have this:

\begin{equation} P\{S_t=k\}=\sum_{n=k}^{\infty}P\{S_t=k | N_t=n\}P\{N_t=n\} \end{equation}

And this is where the issues start to appear because $P\{S_t=k | N_t=n\}$ implicates knowing the distribution of the sum of $n$ independent $LogNormal(\mu,\sigma^2)$ random varibles and, as mentioned by Glen_b in https://stats.stackexchange.com/q/238566 and by the next answer (in descendant order of upvotes), the resulting distribution is a quite hard one to work with and even harder considering that I would still need to find the product of it with my poisson density and -possibly- reduct it to an expression from where I can obtain an inverse function.

Not that I don't find it interesting to follow the above mentioned path but I would rather either knowing a better way to simulate this process or something that helps me make the path that I suggested easier.

Please, help would be appreciated.

Second Idea

A new idea came to my mind, I could try to:

1.- Simulate a set $\{N_i\}_i$ of values from the underlying Poisson Process using that $N_t \sim Poisson(\lambda t)$ (based on the independence of it and the amounts $X_i$),

2.- Simulate $N_i$ values (for each $N_i$ obtained) from a $LogNormal(\mu,\sigma^2)$ random variable (based on the independence among the $X_i$),

3.- Sum the $N_i$ values obtained in 2 for each $N_i$ obtained in 1.

In the end, I would have $|\{N_i\}_i|$ (cardinality) simulations from the Compound Poisson Process.

All of this actually using the Inverse Transform Technique.

Am I right about this?


1 Answer 1


The second idea is the right one. I've already used it.


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