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Rare-Event Simulation for Markov-Modulated Perpetuities: Theory and Algorithms

Dive into the world of perpetuities with a focus on rare-event simulations, examining state-dependent algorithms and novel techniques. Explore first passage problems, theoretical performance, and efficiency, with numerical examples and insightful appendices.

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Rare-Event Simulation for Markov-Modulated Perpetuities: Theory and Algorithms

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  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 3 4 2 Time = 0 1 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 State-Dependent Importance Sampler

  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

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