Multilevel Monte Carlo Metamodeling Imry Rosenbaum Jeremy Staum

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

Simulation Metamodelling

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.

Approaches Regression Interpolation Kriging – Stochastic Kriging Kernel Smoothing

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.

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.

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.

Example

Example Continued

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.

Example Continued

level0L Square root of variance Cost

Generalization

General Thm

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?

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.

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

Goal

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.

The framework

Framework Continued…

Behavior of Adjacnt Points

Multi Level Monte Carlo

Approximating the Response

MLMC Decomposition

MLMC Decomposition Continued

The MLMC estimator

Multilevel Illustration

Multi Level MC estimators

Approximation Reqierments

Bias and Variance of the Approximation

Computational Complexity Theorem

Theorem Continued…

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

Simplifying Assumptions

Simplifying Assumptions Continued

The algorithm

Black-Scholes

Black-Scholes continued

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.

Questions?