In classical risk/ ruin theory, I see this formula crop up in my notes but my lecturer didn't explain to me why/ when it's employed:

$M_X(r) = \int_{-\infty}^{\infty} e^{rx} f(x) dx$

I understand that $M_X(r)$ is a part of the equation: $\lambda M_X(r) - \lambda - cr = 0$,

but for what distribution does this moment generating function apply to?

Here the surplus process is defined as: $U(t) = u + ct - S(t)$, where u is the initial surplus, c is the rate of premium income received, t is the time unit and $S(t)$ is the aggregate claims paid. $S(t) = \sum_{i=1}^{N(t)}X_i$ where $N(t)$ is a counting process distributed under the Poisson distribution with parameter $\lambda$.


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