Adjustment coefficient problem Claims arrive at an insurance company as a Poisson process {$N(t) : t \ge 0$} at rate $\lambda > 0$ and $X_i$ is the claim size of the $ith$ claim.  I assume that {$X_i, i=1,2,...$} is iid (identically and independently distributed) sequence of positive values.  {$X_i, i=1,2,...$} is independent of {$N(t) : t \ge 0$ $S(t)$ is the aggregate loss or total amount of claims to the insurance company.  
I let $t=k$ and I'm supposed to show that $t=k$ is the smallest positive solution to the equation $1+(1 + \theta) \mu t = E[e^{tx}]$, where $\theta$ is a positive constant, called the relative security loading.  If $k$ exists, what is $k$?  I have never seen a problem like this.  By seeing $E[e^{tx}]$, I have a feeling I need to use MGF.  Any suggestions?  Thanks in advance!

Here, $k$ is the Adjustment coefficient, denoted $R$...... you can study this in Ruin Theory.
Yes! $E[e^{Rx}]$ is the MGF of IID RV $X$. But you'd know distribution of $X$.
after getting $R$... you can find upper bound of Probability of Ultimate Ruin by Lundberg Inequality $e^{-RU}$.............. where $U$ is the initial surplus.
