1. Multilevel Monte Carlo Metamodeling Imry Rosenbaum Jeremy Staum

2. Outline What is simulation metamodeling? Metamodeling approaches Why use function approximation? Multilevel Monte Carlo MLMC in metamodeling

3. Simulation Metamodelling

4. Why do we need Metamodeling What-if analysis – How things will change for different scenarios. – Applicable in financial, business and military settings. For example – Multi-product asset portfolios. – How product mix will change our business profit.

5. Approaches Regression Interpolation Kriging – Stochastic Kriging Kernel Smoothing

6. Metamodeling as Function Approximation Metamodeling is essentially function approximation under uncertainty. Information Based Complexity has answers for such settings. One of those answers is Multilevel Monte Carlo.

7. Multilevel Monte Carlo Multilevel Monte Carlo has been suggested as a numerical method for parametric integration. Later the notion was extended to SDEs. In our work we extend the multilevel notion to stochastic simulation metamodeling.

8. Multilevel Monte Carlo In 1998 Stefan Heinrich introduced the notion of multilevel MC. The scheme reduces the computational cost of estimating a family of integrals. We use the smoothness of the underlying function in order to enhance our estimate of the integral.

9. Example

10. Example Continued

13. level0L Square root of variance Cost The variance reaches its maximum in the first level but the cost reaches its maximum in the last level.

14. Example Continued

15. level0L Square root of variance Cost

16. Generalization

17. General Thm

18. Issues MLMC requires smoothness to work, but can we guarantee such smoothness? Moreover, the more dimensions we have the more smoothness that we will require. Is there a setting that will help with alleviating these concerns?

19. Answer The answer to our question came from the derivative estimation setting in Monte Carlo simulation. Derivative Estimation is mainly used in finance to estimate the Greeks of financial derivatives. Glasserman and Broadie presented a framework under which a pathwise estimator is unbiased. This framework will be suitable as well in our case.

20. Simulation MLMC Goal Framework Multi Level Monte Carlo Method Computational Complexity Algorithm Results

21. Goal

22. Elements We will Need for the MLMC Smoothness provided us with the information how adjacent points behave. Our assumptions on the function will provide the same information. The choice of approximation and grid will allow to preserve this properties in the estimator.

23. The framework

24. Framework Continued…

25. Behavior of Adjacnt Points

26. Multi Level Monte Carlo

27. Approximating the Response

28. MLMC Decomposition

29. MLMC Decomposition Continued

30. The MLMC estimator

31. Multilevel Illustration

32. Multi Level MC estimators

33. Approximation Reqierments

34. Bias and Variance of the Approximation

35. Computational Complexity Theorem

36. Theorem Continued…

37. Multilevel Monte Carlo Algorithm The theoretical results need translation into practical settings. Out of simplicity we consider only the Lipschitz continuous setting.

38. Simplifying Assumptions

39. Simplifying Assumptions Continued

40. The algorithm

41. Black-Scholes

42. Black-Scholes continued

43. Conclusion Multilevel Monte Carlo provides an efficient metamodeling scheme. We eliminated the necessity for increased smoothness when dimension increase. Introduced a practical MLMC algorithm for stochastic simulation metamodeling.

44. Questions?