A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

Abstract

Let us consider a risk process with reserve-dependent premium rate and delayed claims. Consider a class of risk processes derived from the original one via scaling in a slow Markov walk sense. In this paper we prove sample path large deviations for the class of risk processes. As a consequence, we give exact asymptotics for the logarithm of the ruin probabilities and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of the ruin probability

Risk processes with shot noise Cox claim number process and reserve dependent premium rate

“…Proof. It is an easy consequence of Proposition 3.3, and it is similar to the proof of Proposition 3.1 of Ganesh et al (2007).…”

“…Risk processes with reservedependent premium rate are quite common in the literature: see e.g. Gerber (1979), Djehiche (1993), Asmussen and Nielsen (1995), Asmussen (2000) and Ganesh et al (2007).…”

“…On the other hand, to model risk processes with delay in claim settlement and reserve-dependent premium rate, Ganesh et al (2007) considered (R(t)) defined by (2) with (S(t)) being a Poisson shot noise process.…”

Applications of Mogulskii, and Kurtz-Feng Large Deviation Results to Risk Reserve Processes with Aggregate Claims

“…The LDP for (reserve dependent premium with delayed claims) risk process was studied by Ganesh, Massi and Torrisi (2007) [7,8]. They proved the LDP with respect to the uniform topology in the case of superexponential claims i.e., claims for which the moment generating function is finite for every 0   .…”

“…They proved the LDP with respect to the uniform topology in the case of superexponential claims i.e., claims for which the moment generating function is finite for every 0   . Later, in [7], they illustrated the connection between risk processes and queues. They applied their large deviations result (valid only in the case of super-exponential claims) to obtain an approximation for the probability of ruin and to propose an importance sampling parameter for simulation.…”