Importance sampling for actuarial cost analysis under a heavy traffic model

Abstract

We explore a bottom-up approach to revisit the problem of cash flow modeling in insurance business, and propose a methodology to efficiently simulate the related tail quantities, namely the fixed-time and the finitehorizon ruin probabilities. Our model builds upon the micro-level contract structure issued by the insurer, and aims to capture the bankruptcy risk exhibited by the aggregation of policyholders. This distinguishes from traditional risk theory that uses random-walk-type model, and also enhances risk evaluation in actuarial pricing practice by incorporating the dynamic arrivals of policyholders in emerging cost analysis. The simulation methodology relies on our model's connection to infinite-server queues with non-homogeneous cost under heavy traffic. We will construct a sequential importance sampler with provable efficiency, along with large deviations asymptotics.

Rare-Event Simulation for Many-Server Queues

“…where h(y) = f (y)/F (y) is the hazard rate function (with the convention that h (y) = ∞ whenever F (y) = 0). In particular, (8) implies that for any p > 0 we can find a > 0 such that yh(y) > p as long as y > a. Hence,F (y) = e − y 0 h(u)du ≤ c 1 e − y a p u du = c 2 y p (9) for some c 1 , c 2 > 0. In other words,F (·) decays faster than any power law.…”

“…where h(y) = f (y)/ F (y) is the hazard rate function (with the convention that h (y) = ∞ whenever F (y) = 0). In particular, (8) implies that for any p > 0 we can find a > 0 such that yh(y) > p as long as y > a. Hence,…”

“…Let V k be r.v. with distribution function F (•) satisfying the light-tail assumption in (8). For any p > 0, we have…”

“…Let V k be r.v. with distribution function F (·) satisfying the light-tail assumption in(8). For any p > 0, we haveE max k=1,...,n V k p = O(l p (n) p ) = o(n ) where l p (n) = inf{y : np ∞ y u p−1F (u)du < η}(47)for a constant η > 0 and is any positive number.Proof.…”

Importance sampling and its optimality for stochastic simulation models

“…Among VRTs, importance sampling (Kahn and Marshall, 1953) is known to be one of the most effective methods and has been used widely in various applications such as communication systems (Heidelberger, 1995;Bucklew, 2004), finance (Owen and Zhou, 2000;Glasserman and Li, 2005), insurance (Blanchet and Lam, 2011), and reliability engineering (Au and Beck, 1999;Lawrence et al, 2013;Choe et al, 2016) to name a few.…”