Importance sampling techniques for the multidimensional ruin problem for general Markov additive sequences of random vectors

• 成果类型: Article
• 署名作者: Collamore, JF
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2002
• 页码: 382-421
• 关键词: large deviations; asymptotic evaluation; process expectations; large time; monte-carlo; Lower bounds; chains; probabilities; functionals; simulation

摘要

Let {(X-n, S-n):n = 0, 1,...} be a Markov additive process, where {x(n)} is a Markov chain on a general state space and S-n is an additive component on R-d. We consider P{S-n is an element of A/epsilon, some n} as epsilon --> 0, where A subset of R-d is open and the mean drift of {S-n} is away from A. Our main objective is to study the simulation of P{S-n is an element of A/epsilon, some n} using the Monte Carlo technique of importance sampling, If the set A is convex, then we establish (i) the precise dependence (as epsilon --> 0) of the estimator variance on the choice of the simulation distribution and (ii) the existence of a unique simulation distribution which is efficient and optimal in the asymptotic sense of D. Siegmund [Ann. Statist. 4 (1976) 673-684] We then extend our techniques to the case where A is not convex. Our results lead to positive conclusions which complement the multidimensional counterexamples of P. Glasserman and Y. Wang [Ann, Appl. Probab. 7 (1997) 731-746].