Rare-Event Simulation for Markov-Modulated Perpetuities Henry Lam Joint Work with Jose Blanchet and Bert Zwart

What is Perpetuity Infinite discounted sum of cash flows Discount rate           Time = 0 1 2 3 4   Cash flow          

Large Deviations Problem  

Assumptions  

First Passage Problem: Light vs Heavy Tail  

First Passage Problem: Simulation  

Tail Behavior of Perpetuity  

Naïve Exponential Tilting  

Key Ideas  

A More General Asymptotic  

A More General Asymptotic   Control on-off of IS

State-Dependent Importance Sampler  

Algorithm  

Markov Modulation  

Algorithm  

Theoretical Performance  

Logarithmic Efficiency  

Finite Termination and Running Time Analysis  

Numerical Example: ARCH(1)  

Numerical Example: ARCH(1) Crude Monte Carlo Estimate C.V. 95% C. I. State-Dependent Importance Sampler Estimate C.V. 95% C. I.

Concluding Remarks A problem with both light and heavy tail behavior Counter example in which naïve exponential tilting fails Novel use of Lyapunov inequality for analysis of state-dependent algorithm

Appendix 1: Efficiency  

Appendix 2: Finite Termination and Running Time Analysis  

Appendix 2: Finite Termination and Running Time Analysis                   Termination