# Simulation of compound Poisson Process with Lognormal jumps?

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:

$$$$S_t=\sum_{i=0}^{N_t}X_i$$$$

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:

$$$$P\{S_t=k\}=\sum_{n=k}^{\infty}P\{S_t=k | N_t=n\}P\{N_t=n\}$$$$

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.

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.