Rate of Convergence of the Probability of Ruin in the Cram\'er-Lundberg Model to its Diffusion Approximation

We analyze the probability of ruin for the {\it scaled} classical Cram\'er-Lundberg (CL) risk process and the corresponding diffusion approximation. The scaling, introduced by Iglehart \cite{I1969} to the actuarial literature, amounts to multiplying the Poisson rate $\la$ by $n$, dividing the claim severity by $\sqrtn$, and adjusting the premium rate so that net premium income remains constant. %Therefore, we think of the associated diffusion approximation as being "asymptotic for large values of $\la$." We are the first to use a comparison method to prove convergence of the probability of ruin for the scaled CL process and to derive the rate of convergence. Specifically, we prove a comparison lemma for the corresponding integro-differential equation and use this comparison lemma to prove that the probability of ruin for the scaled CL process converges to the probability of ruin for the limiting diffusion process. Moreover, we show that the rate of convergence for the ruin probability is of order $\mO\big(n^{-1/2}\big)$, and we show that the convergence is {\it uniform} with respect to the surplus. To the best of our knowledge, this is the first rate of convergence achieved for these ruin probabilities, and we show that it is the tightest one in the general case. For the case of exponentially-distributed claims, we are able to improve the approximation arising from the diffusion, attaining a uniform $\mO\big(n^{-k/2}\big)$ rate of convergence for arbitrary $k \in \N$. We also include two examples that illustrate our results.