Maximal aggregate loss in Risk Process

I am studying the Crámer Lundberg Risk Process. This process is defined as follows: $$Y_t=r_0+ct-Z_t$$ Where $Z_t$ is a compound Poisson sum, it means that $Z_t=\displaystyle\sum_{i=0}^{N_t} X_i$, where $N_t$ is a Poisson process, and $X_t$ are i.i.d random variables that follows a distribution $F$.

Then is defined the maximal aggregate loss, that is: $$S=\sup_{t\geq 0}\{Z_t-ct\}$$

I am trying to simulate this kind of risk process, but I am not clear with the supremum. For that reason, I have been looking some extra information on the maximal aggregate loss. I found in a pdf (file:///C:/Users/lenovo/Downloads/LefevreTrufinZuyderhoff2017_Diffusion.pdf) in page 4 the next expression: That is an equivalent way to compute the maximal aggregate loss that I am interested in. However, I can't see how to prove it.

I think that using this equivalence in R will be easier for me because I need to compute the maximum, in the first expression with the supremum is not so direct because I know that the supremum is the smaller superior bound, it means that the supremum can´t be in the considered set.

Any help with the proof of the equality?

• You might fix the Lefevre Trufin pdf link from your local file to dipot.ulb.ac.be/dspace/bitstream/2013/258727/3/… – javadba Jan 8 '18 at 1:41
• if you're asking for a proof of the equality, you might want to define the $T_i$ – Taylor Jan 8 '18 at 3:43
• $T_i$ is the time between two consecutivo arrivals, under this framework is an exponential random variable, since is a compound poisson process. – Boris Jan 8 '18 at 4:16