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

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

3. Large Deviations Problem

4. Assumptions

5. First Passage Problem: Light vs Heavy Tail

6. First Passage Problem: Simulation

7. Tail Behavior of Perpetuity

8. Naïve Exponential Tilting

9. Key Ideas

10. A More General Asymptotic

11. A More General Asymptotic
Control on-off of IS

12. State-Dependent Importance Sampler

13. Algorithm

14. Markov Modulation

15. Algorithm

16. Theoretical Performance

17. Logarithmic Efficiency

18. Finite Termination and Running Time Analysis

19. Numerical Example: ARCH(1)

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

21. 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

22. Appendix 1: Efficiency

23. Appendix 2: Finite Termination and Running Time Analysis

24. Appendix 2: Finite Termination and Running Time Analysis
Termination