Rare event simulation for processes generated via stochastic fixed point equations

TY - JOUR

T1 - Rare event simulation for processes generated via stochastic fixed point equations

AU - Collamore, Jeffrey F.

AU - Diao, Guoqing

AU - Vidyashankar, Anand N.

PY - 2014/10

Y1 - 2014/10

N2 - In a number of applications, particularly in financial and actuarial mathematics, it is of interest to characterize the tail distribution of a random variable V satisfying the distributional equation V D= f (V), where f (v) = Amax{v,D}+B for (A,B,D) (0,∞)×ℝ 2. This paper is concerned with computational methods for evaluating these tail probabilities. We introduce a novel importance sampling algorithm, involving an exponential shift over a random time interval, for estimating these rare event probabilities. We prove that the proposed estimator is: (i) consistent, (ii) strongly efficient and (iii) optimal within a wide class of dynamic importance sampling estimators. Moreover, using extensions of ideas from nonlinear renewal theory, we provide a precise description of the running time of the algorithm. To establish these results, we develop new techniques concerning the convergence of moments of stopped perpetuity sequences, and the first entrance and last exit times of associated Markov chains on R. We illustrate our methods with a variety of numerical examples which demonstrate the ease and scope of the implementation.

AB - In a number of applications, particularly in financial and actuarial mathematics, it is of interest to characterize the tail distribution of a random variable V satisfying the distributional equation V D= f (V), where f (v) = Amax{v,D}+B for (A,B,D) (0,∞)×ℝ 2. This paper is concerned with computational methods for evaluating these tail probabilities. We introduce a novel importance sampling algorithm, involving an exponential shift over a random time interval, for estimating these rare event probabilities. We prove that the proposed estimator is: (i) consistent, (ii) strongly efficient and (iii) optimal within a wide class of dynamic importance sampling estimators. Moreover, using extensions of ideas from nonlinear renewal theory, we provide a precise description of the running time of the algorithm. To establish these results, we develop new techniques concerning the convergence of moments of stopped perpetuity sequences, and the first entrance and last exit times of associated Markov chains on R. We illustrate our methods with a variety of numerical examples which demonstrate the ease and scope of the implementation.

U2 - 10.1214/13-AAP974

DO - 10.1214/13-AAP974

M3 - Journal article

SN - 1050-5164

VL - 24

SP - 2143

EP - 2175

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 5

ER -