# Classical Ruin Theory - Lundberg Model

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 $$\lambda$$.